# SSAT Upper Level Math : How to find the probability of an outcome

## Example Questions

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### Example Question #1681 : Ssat Upper Level Quantitative (Math)

Eight standard decks of 52 cards are shuffled together; four jokers are shuffled in as well. A card is selected at random from this new deck. What is the probability that a spade will be selected?

Explanation:

Eight decks of 52 cards, with 4 jokers added, comprise a total of

cards. A standard deck of 52 cards has 13 spades, so the eight decks together have a total of

spades. This makes the probability of a random draw resulting in a spade

### Example Question #1681 : Ssat Upper Level Quantitative (Math)

The above set of balls is placed in a box. Give the probability that, if a ball is drawn at random, it will be a gray ball marked with an odd number.

Explanation:

As seen in the diagram below, nine of the twenty-five balls in the box are both gray and marked with odd numbers.

This makes the probability that a random selection will yield one of these balls  .

### Example Question #31 : Outcomes

The above set of balls is placed in a box. Give the probability that, if a ball is drawn at random, it will be a white ball marked with a prime number.

Explanation:

Out of the whole numbers from 1 through 9, there are four prime numbers - 2, 3, 5, and 7. As seen in the diagram below, there are five out of twenty-five balls that are both white and marked with a prime number.

This makes the probability of drawing one of these balls .

### Example Question #55 : Data Analysis / Probablility

Mark is playing a board game in which spaces represent property that can be "bought" and "sold"; the above diagram represents part of the board. Mark is represented by the red triangle, and he is moving in a left-to-right direction.

Mark is about to roll a pair of fair six-sided dice. If he lands on one of the yellow spaces, he will have to pay a huge fee to his opponent, Jennifer, who "owns" the properties represented by the spaces; if this happens, he will be out of the game.

What are the odds of this happening?

11 to 7 against

2 to 1 against

5 to 4 against

Even odds

11 to 7 against

Explanation:

The undesirable rolls are 5, 6, and 8. As can be seen in the diagram below, out of thirty-six equally probable outcomes possible with the roll of two fair six-sided dice, fourteen are undesirable, and twenty-two are desirable.

The odds against an undesirable roll are

- that is, 11 to 7.

### Example Question #31 : Probability

The above set of balls is placed in a box. Give the probability that, if a ball is drawn at random, it will be a gray ball marked with a number that is a multiple of 3.

Explanation:

Of the twenty-five balls in the box, five are both gray and marked with a number that is a multiple of 3 (3, 6, 90), as seen in the figure below:

This makes the probability of drawing one of these balls

.

### Example Question #32 : Probability

Above is a part of the board for a popular board game. Terry is moving clockwise on the board; his playing piece is represented by the black triangle. He is about to roll a pair of fair six-sided dice. He collects $10,000 every time he lands on or passes the space marked "Onward!" What are the odds in favor of, or against, him collecting$10,000 on this turn?

11 to 1 in favor

11 to 1 against

5 to 1 against

5 to 1 in favor

11 to 1 in favor

Explanation:

Terry will collect $10,000 if he rolls a four or higher. As can be seen in the diagram below, out of thirty-six equally probable outcomes possible with the roll of two fair six-sided dice, there are only three rolls unfavorable to this outcome, and thirty-three favorable: This makes the odds , or 11 to 1, in favor of$10,000 being collected on this turn.

### Example Question #32 : Probability

In the game of Crazy Eights, which is played with an ordinary deck of 52 playing cards, each player lays down a card from his hand on top of a discard pile according to the card that has previously been played. The card he is allowed to play must be a card of the same rank or suit, or it must be an eight of any suit.

At the beginning of the hand, the seven of diamonds was turned over; Kenny, who was first, played a six of diamonds on top of it. Dick, who plays next, has the following cards in his hand:

As can be seen, Dick does not have a playable card. By the rules, he must draw a card, and, if he can play it, he must. What is the probability that the card he draws will be playable?

Explanation:

Dick knows 10 of the cards the deck - the eight in his hand, and the two in the discard pile. Of the other  cards, the following are playable:

The four eights;

The other three sixes (hearts, clubs, spades); and,

The other ten diamonds (the six and the seven have been played, and the eight of diamonds has already been counted).

This makes

of the 42 cards playable, so the probability that a playable card will be drawn is .

### Example Question #33 : Probability

In the game of Crazy Eights, which is played with an ordinary deck of 52 playing cards, each player lays down a card from his hand on top of a discard pile according to the card that has previously been played. The card he is allowed to play must be a card of the same rank or suit, or it must be an eight of any suit.

At the beginning of the hand, the two of clubs has been turned over. Brenda has in her hand eight cards, as follows:

As can be seen, Brenda cannot play a card. By the rules, she must draw a card, and, if she can play it, she must. What is the probability that the card she draws will be playable?

Explanation:

Brenda knows nine of the cards - the one in the discard pile and the eight in her hand. Therefore, the next card can be one of  different cards. The playable cards are:

The four eights;

The other three twos (hearts, spades, diamonds); and,

The other eleven clubs (the two has been played, and the eight of diamonds has already been counted).

This makes

of the 42 cards playable, so the probability that a playable card will be drawn is .

### Example Question #1691 : Ssat Upper Level Quantitative (Math)

In a jar, there are 3 blue marbles, 5 red marbles, 8 green marbles. If Bob reaches his hand in a jar, and grabs one marble, what is the likelihood he will pick up a blue marble?

Explanation:

First, calculate the total number of marbles in the jar, which is . Because 3 of the marbles are blue, the chances of picking a blue marble .

### Example Question #1696 : Ssat Upper Level Quantitative (Math)

A fair six-sided die has on its faces the numbers 1, 1, 1, 2, 2, 3; a fair eight-sided die has on its faces the numbers 1, 1, 1, 1, 2, 2, 3, 4. If both dice are rolled, what is the probability that the numbers add up to 3 or less?

Explanation:

In order for the roll to add up to 3 or less, neither die can come up 3 or 4. Therefore, we look at the probabilities that each die can come up 1 and 2.

The six-sided die comes up 1 with probability ; it comes up 2 with probability .

The six-sided die comes up 1 with probability ; it comes up 2 with probability .

A roll of 3 or less can happen three ways:

Case 1: Both dice come up 1. The probability of this, by the Multiplication Principle, is

Case 2: The six-sided die comes up 1 and the eight-sided die comes up 2. The probability of this, by the Multiplication Principle, is

Case 3: The six-sided die comes up 2 and the eight-sided die comes up 1. The probability of this, by the Multiplication Principle, is

These are mutually exclusive events, so add these probabilities:

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