How to find the probability of an outcome - ISEE Middle Level Quantitative Reasoning
Card 0 of 550
Two standard decks of fifty-two cards, one with a red backing and one with a purple backing, are presented to you. The red deck has all of its aces removed; the purple deck has all of its kings removed. Which is the greater quantity?
(A) The probability that a card randomly drawn from the red deck is a seven
(B) The probability that a card randomly drawn from the purple deck is a seven
Two standard decks of fifty-two cards, one with a red backing and one with a purple backing, are presented to you. The red deck has all of its aces removed; the purple deck has all of its kings removed. Which is the greater quantity?
(A) The probability that a card randomly drawn from the red deck is a seven
(B) The probability that a card randomly drawn from the purple deck is a seven
Both modified decks still have four sevens, and both modified decks have forty-eight cards. Therefore, the probability of drawing a seven from the red deck is the same as the probability of drawing a seven from the purple deck: 
.
Both modified decks still have four sevens, and both modified decks have forty-eight cards. Therefore, the probability of drawing a seven from the red deck is the same as the probability of drawing a seven from the purple deck:
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Two fair six-sided dice are thrown. What is the probability that the product is greater than or equal to 20?
Two fair six-sided dice are thrown. What is the probability that the product is greater than or equal to 20?
The rolls that yield a product greater than or equal to 20 are:
These are 8 out of 36 rolls, so the probability of getting one of them is 
The rolls that yield a product greater than or equal to 20 are:
These are 8 out of 36 rolls, so the probability of getting one of them is
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Using the information given in each question, compare the quantity in Column A to the quantity in Column B.
A pair of regular dice (with sides numbered from 1-6) is thrown.
Column A Column B
the odds of the odds of
rolling a total of 8 rolling a total of 7
Using the information given in each question, compare the quantity in Column A to the quantity in Column B.
A pair of regular dice (with sides numbered from 1-6) is thrown.
Column A Column B
the odds of the odds of
rolling a total of 8 rolling a total of 7
Consider the different ways in which you could roll an 8 or a 7.
You could roll 8 by the following combinations:
2-6, 3-5, 4-4, 5-3, 6-3
so the odds of rolling 8 are 
.
You could roll 7 by the following combinations:
1-6, 2-5, 3-4, 4-3, 5-2, 6-1
so the odds of rolling 7 are 
.
The answer, therefore, is that Column B is greater.
Consider the different ways in which you could roll an 8 or a 7.
You could roll 8 by the following combinations:
2-6, 3-5, 4-4, 5-3, 6-3
so the odds of rolling 8 are
You could roll 7 by the following combinations:
1-6, 2-5, 3-4, 4-3, 5-2, 6-1
so the odds of rolling 7 are
The answer, therefore, is that Column B is greater.
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A card is drawn at random from a deck of 53 cards - the standard deck including the joker. Which is the greater quantity?
(a) The probability of drawing a black card
(b) 
A card is drawn at random from a deck of 53 cards - the standard deck including the joker. Which is the greater quantity?
(a) The probability of drawing a black card
(b)
26 of the 53 cards are black (the joker counts as neither).
Half of 53 is
26 is less than this, so black cards comprise less than half the deck, and the probability of drawing a black card is less than 
.
26 of the 53 cards are black (the joker counts as neither).
Half of 53 is
26 is less than this, so black cards comprise less than half the deck, and the probability of drawing a black card is less than
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A standard deck of fifty-two cards is altered by removing the tens and replacing them with the queens from another deck. A card is drawn at random from the altered deck.
Which is the greater quantity?
(a) The probability that the card is a face card
(b) 
Note: a face card is a jack, a queen, or a king.
A standard deck of fifty-two cards is altered by removing the tens and replacing them with the queens from another deck. A card is drawn at random from the altered deck.
Which is the greater quantity?
(a) The probability that the card is a face card
(b)
Note: a face card is a jack, a queen, or a king.
With the replacement of the tens with the queens, there are still 52 cards in the deck, but now, there are four jacks, eight queens, and four kings - 16 face cards. The probability that a random card is a face card is
With the replacement of the tens with the queens, there are still 52 cards in the deck, but now, there are four jacks, eight queens, and four kings - 16 face cards. The probability that a random card is a face card is
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A standard deck of fifty-two cards is altered by removing the black aces. A card is drawn at random from the altered deck.
Which is the greater quantity?
(a) The probability that a face card will be drawn
(b) 
Note: a face card is a jack, a queen, or a king.
A standard deck of fifty-two cards is altered by removing the black aces. A card is drawn at random from the altered deck.
Which is the greater quantity?
(a) The probability that a face card will be drawn
(b)
Note: a face card is a jack, a queen, or a king.
The removal of the two black aces leaves a deck of 50 cards, with all 12 face cards remaining. The probability that a randomly drawn card is a face card is therefore
Since 
, the probability is less than 
.
The removal of the two black aces leaves a deck of 50 cards, with all 12 face cards remaining. The probability that a randomly drawn card is a face card is therefore
Since
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Two fair dice are thrown. Which is the greater quantity?
(a) The probability that the sum wiil be 5 or less
(b) 
Two fair dice are thrown. Which is the greater quantity?
(a) The probability that the sum wiil be 5 or less
(b)
For the sum of the dice to be 5 or less, one of the following rolls must be thrown:
(1,1). (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1)
This makes 10 out of 36 rolls. Since one-fourth of 36 is 9, the probability of throwing a 5 or less is greater than 
.
For the sum of the dice to be 5 or less, one of the following rolls must be thrown:
(1,1). (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1)
This makes 10 out of 36 rolls. Since one-fourth of 36 is 9, the probability of throwing a 5 or less is greater than
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Sandy has two red dice; Tommy has one blue die. All three dice are fair.
Sandy rolls her red dice once and notes her sum. Tommy rolls his blue die twice and notes his sum.
Which is the greater quantity?
(a) The probability that Sandy will roll a seven or higher
(b) The probability that Tommy will roll a seven or higher
Sandy has two red dice; Tommy has one blue die. All three dice are fair.
Sandy rolls her red dice once and notes her sum. Tommy rolls his blue die twice and notes his sum.
Which is the greater quantity?
(a) The probability that Sandy will roll a seven or higher
(b) The probability that Tommy will roll a seven or higher
Since each die is fair, the roll of each die is an independent event; also, the second roll of the blue die is independent of the first roll. Therefore, the probabilities of each outcome are the same for Sandy rolling two dice simultaneously as for Tommy rolling one twice. (a) and (b) are equal.
Since each die is fair, the roll of each die is an independent event; also, the second roll of the blue die is independent of the first roll. Therefore, the probabilities of each outcome are the same for Sandy rolling two dice simultaneously as for Tommy rolling one twice. (a) and (b) are equal.
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A standard deck of fifty-two cards is altered by removing the jacks and replacing them with the queens from another deck. A card is drawn at random from the altered deck.
Which is the greater quantity?
(a) The probability that a black card is drawn
(b) The probability that a red card is drawn
A standard deck of fifty-two cards is altered by removing the jacks and replacing them with the queens from another deck. A card is drawn at random from the altered deck.
Which is the greater quantity?
(a) The probability that a black card is drawn
(b) The probability that a red card is drawn
The removal of the jacks removes two black cards and two red cards from the deck; replacement with the four queens from another deck adds two black cards and two red cards. The number of black cards and the number of red cards remain the same, so the probabilities remain equal.
The removal of the jacks removes two black cards and two red cards from the deck; replacement with the four queens from another deck adds two black cards and two red cards. The number of black cards and the number of red cards remain the same, so the probabilities remain equal.
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There is a bag with 10 yellow markers, 3 red markers, 2 blue markers, and 5 green markers. What is the probability of picking a marker that is not red?
There is a bag with 10 yellow markers, 3 red markers, 2 blue markers, and 5 green markers. What is the probability of picking a marker that is not red?
Probability involves part over whole. Therefore, you must find the total number of markers, which is 20. Then, combine all of the markers that are not red 
, which gives you 17. Put 17 over 20 to get 
Probability involves part over whole. Therefore, you must find the total number of markers, which is 20. Then, combine all of the markers that are not red
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Two boxes contain only red and green marbles. Box #1 contains five green marbles and ten red marbles; Box #2 contains ten green marbles and thirty red marbles. Which is the greater quantity?
(A) The probability that a marble randomly drawn from Box #1 is green
(B) The probability that a marble randomly drawn from Box #2 is green
Two boxes contain only red and green marbles. Box #1 contains five green marbles and ten red marbles; Box #2 contains ten green marbles and thirty red marbles. Which is the greater quantity?
(A) The probability that a marble randomly drawn from Box #1 is green
(B) The probability that a marble randomly drawn from Box #2 is green
Box #1 contains five green marbles and fifteen marbles overall; a marble randomly drawn from this box will be green with probability 
Box #2 contains ten green marbles and forty marbles overall; a marble randomly drawn from this box will be green with probability 
,
so (A) is greater.
Box #1 contains five green marbles and fifteen marbles overall; a marble randomly drawn from this box will be green with probability
Box #2 contains ten green marbles and forty marbles overall; a marble randomly drawn from this box will be green with probability
so (A) is greater.
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A box contains fifteen blue marbles, ten green marbles, and twenty red marbles, and no other marbles. What are the odds against a marble randomly drawn from this box being blue?
A box contains fifteen blue marbles, ten green marbles, and twenty red marbles, and no other marbles. What are the odds against a marble randomly drawn from this box being blue?
Since there are 30 marbles that are not blue and 15 that are blue, this makes 30 unfavorable outcomes to 15 favorable outcomes. This makes the odds against this favorable outcome, in fraction form:
,
or 2 to 1 odds.
Since there are 30 marbles that are not blue and 15 that are blue, this makes 30 unfavorable outcomes to 15 favorable outcomes. This makes the odds against this favorable outcome, in fraction form:
or 2 to 1 odds.
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Two standard decks of fifty-two cards, one with a green backing and one with a blue backing, are presented to you. The green deck has all of its hearts removed; the blue deck is left complete. Which is the greater quantity?
(A) The probability that a card randomly drawn from the green deck is a two
(B) The probability that a card randomly drawn from the blue deck is a two
Two standard decks of fifty-two cards, one with a green backing and one with a blue backing, are presented to you. The green deck has all of its hearts removed; the blue deck is left complete. Which is the greater quantity?
(A) The probability that a card randomly drawn from the green deck is a two
(B) The probability that a card randomly drawn from the blue deck is a two
The green deck, having had all of its hearts removed, has thirty-nine cards; one of the hearts removed is a two, so there are three of the four twos left. The probability of a card drawn from the green deck being a two is 
.
The blue deck, being complete, has fifty-two cards, including all four of its twos. The probability of a card drawn from the blue deck being a two is 
.
The two probabilities are equal.
The green deck, having had all of its hearts removed, has thirty-nine cards; one of the hearts removed is a two, so there are three of the four twos left. The probability of a card drawn from the green deck being a two is
The blue deck, being complete, has fifty-two cards, including all four of its twos. The probability of a card drawn from the blue deck being a two is
The two probabilities are equal.
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Two boxes contain only red and green marbles. Box #1 contains eight green marbles and ten red marbles; Box #2 contains twenty green marbles and sixteen red marbles. Which is the greater quantity?
(A) The probability that a marble randomly drawn from Box #1 is green
(B) The probability that a marble randomly drawn from Box #2 is red
Two boxes contain only red and green marbles. Box #1 contains eight green marbles and ten red marbles; Box #2 contains twenty green marbles and sixteen red marbles. Which is the greater quantity?
(A) The probability that a marble randomly drawn from Box #1 is green
(B) The probability that a marble randomly drawn from Box #2 is red
Box #1 has eight green marbles out of eighteen marbles total; the probability that a randomly drawn marble is green is 
.
Box #2 has sixteen red marbles out of thirty-six marbles total; the probability that a randomly drawn marble is green is 
.
The two probabilities are equal.
Box #1 has eight green marbles out of eighteen marbles total; the probability that a randomly drawn marble is green is
Box #2 has sixteen red marbles out of thirty-six marbles total; the probability that a randomly drawn marble is green is
The two probabilities are equal.
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Two boxes contain thirty marbles each: red, blue, and green. Box #1 contains ten green marbles and fifteen red marbles; Box #2 contains sixteen green marbles and nine red marbles. Which is the greater quantity?
(A) The probability that a marble randomly drawn from Box #1 is blue.
(A) The probability that a marble randomly drawn from Box #2 is blue.
Two boxes contain thirty marbles each: red, blue, and green. Box #1 contains ten green marbles and fifteen red marbles; Box #2 contains sixteen green marbles and nine red marbles. Which is the greater quantity?
(A) The probability that a marble randomly drawn from Box #1 is blue.
(A) The probability that a marble randomly drawn from Box #2 is blue.
Box #1 contains 
blue marbles out of 30 total.
Box #2 contains 
blue marbles out of 30 total.
The probability of drawing a blue marble out of Box #1 is the same as that of drawing a blue marble out of Box #2: 
.
Box #1 contains
Box #2 contains
The probability of drawing a blue marble out of Box #1 is the same as that of drawing a blue marble out of Box #2:
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Two standard decks of fifty-two cards, one with a yellow backing and one with a gray backing, are presented to you. The yellow deck has all of its aces removed; the gray deck is left complete. Which is the greater quantity?
(A) The probability that a card randomly drawn from the yellow deck is a six
(B) The probability that a card randomly drawn from the gray deck is a six
Two standard decks of fifty-two cards, one with a yellow backing and one with a gray backing, are presented to you. The yellow deck has all of its aces removed; the gray deck is left complete. Which is the greater quantity?
(A) The probability that a card randomly drawn from the yellow deck is a six
(B) The probability that a card randomly drawn from the gray deck is a six
The yellow deck has four sixes out of forty-eight cards; the probability of drawing a six at random is 
.
The gray deck has four sixes out of fifty-two cards; the probability of drawing a six at random is 
.
Between two fractions with the same numerator, the fraction with the lesser denominator is the greater, so 
. (A) is greater.
The yellow deck has four sixes out of forty-eight cards; the probability of drawing a six at random is
The gray deck has four sixes out of fifty-two cards; the probability of drawing a six at random is
Between two fractions with the same numerator, the fraction with the lesser denominator is the greater, so
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A standard deck of fifty-two cards has all of its hearts removed. A card is then drawn at random. Which is the greater quantity?
(A) The probability that the card is a club.
(B) The probability that the card is a 2, a 3, or a 4.
A standard deck of fifty-two cards has all of its hearts removed. A card is then drawn at random. Which is the greater quantity?
(A) The probability that the card is a club.
(B) The probability that the card is a 2, a 3, or a 4.
We need only compare numbers of cards.
Since no clubs were removed, there are thirteen clubs in the modified deck.
Since one 2, one 3, and one 4 were removed, there are three of each, or nine of these cards total, in the modified deck.
The probability of drawing a club, (A), is greater.
We need only compare numbers of cards.
Since no clubs were removed, there are thirteen clubs in the modified deck.
Since one 2, one 3, and one 4 were removed, there are three of each, or nine of these cards total, in the modified deck.
The probability of drawing a club, (A), is greater.
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A standard deck of fifty-two cards has all of its aces removed. A card is then drawn at random. Which is the greater quantity?
(A) The probability that the card is a diamond.
(B) The probability that the card is a jack, a queen, or a king.
A standard deck of fifty-two cards has all of its aces removed. A card is then drawn at random. Which is the greater quantity?
(A) The probability that the card is a diamond.
(B) The probability that the card is a jack, a queen, or a king.
We need only compare numbers of cards.
Since one diamond (the ace of diamonds) was removed, there are twelve diamonds in the modified deck.
Since no jacks, queens, or kings were removed, all four of each (twelve cards total) remain.
This makes the two quantities and, subsequently, the two probabilities, equal.
We need only compare numbers of cards.
Since one diamond (the ace of diamonds) was removed, there are twelve diamonds in the modified deck.
Since no jacks, queens, or kings were removed, all four of each (twelve cards total) remain.
This makes the two quantities and, subsequently, the two probabilities, equal.
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A box contains fifteen blue marbles, ten green marbles, twenty red marbles, and twenty-five white marbles, and no other marbles. What are the odds against a marble randomly drawn from this box being blue or green?
A box contains fifteen blue marbles, ten green marbles, twenty red marbles, and twenty-five white marbles, and no other marbles. What are the odds against a marble randomly drawn from this box being blue or green?
There are 45 marbles that are neither blue nor green, and twenty-five marbles that are blue or green, so the odds against drawing a blue or green marble are 45 to 25, or
That is, 9 to 5 odds against.
There are 45 marbles that are neither blue nor green, and twenty-five marbles that are blue or green, so the odds against drawing a blue or green marble are 45 to 25, or
That is, 9 to 5 odds against.
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A box contains ten red marbles and fifteen blue marbles. Fifteen more red marbles are added to the box. By how much did the probability that a randomly drawn marble is red increase?
A box contains ten red marbles and fifteen blue marbles. Fifteen more red marbles are added to the box. By how much did the probability that a randomly drawn marble is red increase?
Originally, the box contained twenty-five marbles, ten of which were red, so the probability of drawing a red marble was
After adding fifteen red marbles, the box contains forty marbles, twenty-five of which are red, so the probability of drawing a red marble is now
The increase in probability is
Originally, the box contained twenty-five marbles, ten of which were red, so the probability of drawing a red marble was
After adding fifteen red marbles, the box contains forty marbles, twenty-five of which are red, so the probability of drawing a red marble is now
The increase in probability is
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