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## Example Questions

### Example Question #1 : How To Find The Probability Of An Outcome

A bag of jellybeans has 20 watermelon jellybeans, 45 sour apple jellybeans, 30 orange jellybeans and 5 cotton candy jellybeans. If you reach in and grab one jelly bean, what is the probability that it will be watermelon flavored?

**Possible Answers:**

1/3

1/4

1/5

1/20

4/19

**Correct answer:**

1/5

Add up the total number of jellybeans, 20 + 45 + 30 + 5 = 100.

Divide the number of watermelon jellybeans by the total: 20/100 and reduce the fraction to 1/5.

### Example Question #1 : How To Find The Probability Of An Outcome

A circle is inscribed inside a square. If a point inside the square is selected at random, what is the probability that the point will also be inside the circle?

**Possible Answers:**

3/4

5/6

π/6

π/4

**Correct answer:**

π/4

The probability of the point being inside the circle is the ratio of the area of the circle to the area of the square. If we suppose that the circle has radius r, then the square must have side 2r. The area of the circle is πr^{2} and the area of the square is 〖(2r)〗^{2}=〖4r〗^{2}, so the proportion of the areas is (πr^{2})/〖4r〗^{2} =π/4.

### Example Question #1 : How To Find The Probability Of An Outcome

John has a bowl with 54 marbles in it. Half of the marbles are green and half of the marbles are blue. John takes 3 green marbles and 6 blue marbles from the jar. John then takes 13 additional marbles from the remaining marbles in the jar. What is the minimum number of these 13 marbles that must be green in order for there to be more blue marbles than green marbles left?

**Possible Answers:**

10

8

4

5

9

**Correct answer:**

9

The bowl has 54 marbles, half green and half blue. This gives us 27 green and 27 blue marbles:

27 G / 27 B

John then takes 3 green and 6 blue from the bowl. This leaves the bowl with:

24 G / 21 B

If there are going to be more blue than green marbles after John's 13 marbles, he has to take at least 4 more green marbles than blue marbles, because right now there are 3 less blue marbles. Therefore, we need to take **at least 9 green marbles**, which would mean 4 or less of the marbles would be blue (8 green and 5 blue would leave us with equal green and equal blue marbles, so it would have to be more than 8 green marbles, which gives us 9 green marbles).

We can also solve this as an inequality. You take the difference in marbles, which is 3, which means you need the difference in green and blue marbles to be greater than 3, or at least 4. You have b + g = 13 and g - b > 3, where b and g are positive integers.

b + g = 13 (Subtract g on both sides of the equation)

b = 13 - g

g - b > 3 (Substitute above equation)

g - (13 - g) > 3 (Distribute negative sign in parentheses)

g - 13 + g > 3 (Add both g variables)

2g - 13 > 3 (Add 13 to both sides of the inequality)

2g > 16 (Divide both sides of the inequality by 2)

g > 8 so g has to be 9 or greater.

### Example Question #1 : How To Find The Probability Of An Outcome

If x is chosen at random from the set (4, 6, 7, 9, 11) and y is chosen at random from the set (12, 13, 15, 17) then what is the probability that xy is odd?

**Possible Answers:**

9/10

9/20

6/10

11/20

3/10

**Correct answer:**

9/20

If x is chosen at random from the set (4, 6, 7, 9, 11) and y is chosen at random from the set (12, 13, 15, 17) then what is the probability that xy is odd?

Here we have 5 possible choices for x and 4 possible choices for y, giving us 5 * 4 = 20 possible outcomes.

We know that odd times odd = odd; even times even = even; and even times odd = even. Thus we need all of the outcomes where x and y are odd. We have 3 possibilities of odd numbers for x, and 3 possibilities of odd numbers for y, so we will have 9 outcomes of our total 20 outcomes where xy is odd, giving us a probability of **9/20**.

### Example Question #1 : How To Find The Probability Of An Outcome

Mike has a bag of marbles, 4 white, 8 blue, and 6 red. He pulls out one marble from the bag and it is red. What is the probability that the second marble he pulls out of the bag is white?

**Possible Answers:**

4/17

4/18

1/6

3/18

**Correct answer:**

4/17

There are 18 marbles in total. One of them is removed so now there are 17 marbles. This is our denominator. All of the original white marbles are still in the bag so there is a 4 out of 17 or 4/17 chance that the next marble taken out of the bag will be white.

### Example Question #1 : How To Find The Probability Of An Outcome

Michael tosses three fair coins. What is the probability that at least one of these coins will land on heads?

**Possible Answers:**

3/4

7/8

1/2

1/8

1/4

**Correct answer:**

7/8

Michael can toss either one head, two heads, or three heads.

If Michael tosses one head, then it could be on either the first, second, or third toss. We could model it like this, where H represents heads and T represents tails.

HTT, THT, or TTH

If Michael tosses two heads, then there are three possible combinations:

HHT, HTH, or THH

If Michael tosses three heads, then there is only one possible combination:

HHH

Thus, there are seven ways that Michael can toss at least one head. We must find the probability of each of these ways and then add them together.

The probability of rolling a head is ½ and the probability of rolling a tail is ½. Because each coin toss is independent, we can multiply the probabilities together.

For example, the probability of the combination HTT is (1/2)(1/2)(1/2) = 1/8

Probability of HTT = 1/8

Probability of THT = (1/2)(1/2)(1/2) = 1/8

Probability of TTH = (1/2)(1/2)(1/2) = 1/8

Probability of HHT = 1/8

Probability of HTH = 1/8

Probability of THH = 1/8

Probability of HHH = 1/8

So, there are seven possible ways that Michael can toss at least one head. The probability of each of these seven ways is equal to 1/8. Thus, the total probability of all seven events is 7/8.

ALTERNATE SOLUTION:

Michael can toss at least one head, or he can toss zero heads. The sum of these two probabilities must equal one, because they represent all of the ways that Michael could toss the coins. He could either toss at least on head, or he could toss no heads at all.

Probability of tossing at least one head + probability of tossing no heads = 1

The probability of tossing no heads is only possible with the combination TTT. The probability of tossing three tails is equal to (1/2)(1/2)(1/2) = 1/8

Probability of tossing at least one head + 1/8 = 1

Probability of tossing at least one head = 1 – 1/8 = 7/8 .

### Example Question #7 : How To Find The Probability Of An Outcome

A jar of marbles has 125 marbles in it. 25 are blue marbles, 65 are red marble, 15 are green marbles and 20 are yellow marbles. What is the probability that the first three marbles picked are green or blue?

**Possible Answers:**

0.025

0.043

0.031

0.020

0.015

**Correct answer:**

0.031

Probability of each event = (# green marbles + # blue marbles)/ Total # of Marbles

P1 = (15 + 25) / 125 = 40 / 125

Second event assumes a blue or green was chosen for first event so there is one fewer marble on top and also one fewer marble in the total number of marbles.

P2 = (14 + 25) / 124 = 39 / 124

Third event assumes a blue or green was chosen for first and second events so there are two fewer marbles on top and also two fewer marbles in the total number of marbles.

P3 = (13 + 25) / 124 = 38 / 123

Probability for multiple events = P1 x P2 x P3

(40 / 125) * (39 / 124) * (38 / 123)

( 40 * 39 * 38) / (125 * 124 * 123 ) = 59280 / 1906500 = 0.031

### Example Question #2 : How To Find The Probability Of An Outcome

If given two dice, what is the probability that the sum of the two numbers rolled will equal 9?

**Possible Answers:**

1/9

1/18

1/6

1/24

1/36

**Correct answer:**

1/9

There are 36 possible outcomes of the additive dice roll. The way to roll a sum of 9 is 6 (and vice versa) and 3 or 5 and 4 (and vice versa). This is possible 4 of the 36 times, giving a probability the sum of the two dice rolled of 4/36 or 1/9.

### Example Question #9 : How To Find The Probability Of An Outcome

A bag contains 6 green marbles, 5 blue, and 9 red. What is the probability that you will select two green marbles from the bag?

**Possible Answers:**

6/20

5/42

3/38

21/190

9/100

**Correct answer:**

3/38

There are 20 total marbles. Selecting the first green marble has a 6/20 chance, the second green marble has a 5/19 chance. This gives a total chance of 30/380, or a 3/38 chance.

### Example Question #2 : How To Find The Probability Of An Outcome

There is a special contest held at a high school where the winner will receive a prize of $100. 300 seniors, 200 juniors, 200 sophomores, and 100 freshmen enter the contest. Each senior places their name in the hat 5 times, juniors 3 times, and sophmores and freshmen each only once. What is the probability that a junior's name will be chosen?

**Possible Answers:**

1/6

1/24

2/5

1/4

5/8

**Correct answer:**

1/4

The first thing to do here is find the total number of students in the contest. Seniors = 300 * 5 = 1500, Juniors = 200 * 3 = 600, Sophomores = 200, and Freshmen = 100. So adding all these up you get a total of 2400 names in the hat. Out of these 2400 names, 600 of them are Juniors. So the probability of choosing a Junior's name is 600/2400 = 1/4.

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