How to use the multiplication rule

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AP Statistics › How to use the multiplication rule

Questions 1 - 10
1

If you have a deck of cards, what is the probability that you draw a spade after you drew a non-spade on the first draw without replacement?

Explanation

You must use the multiplication rule which is the probability of one event happening after one has already taken place is the product of both probabilities. The probability of drawing a non-spade on the first draw is . Since there is no replacment, there are now 51 cards in the deck. The probability of drawing the spade on the second draw is . The probability of both happening after one another is then =

2

With a standard deck of cards, what is the probability of picking a spade then a red card if there is no replacement?

Explanation

In a standard deck of cards:

3

In a standard deck of cards, without replacement, what is the probability of drawing three kings?

Explanation

Start with 52 cards, probability of drawing first king:

Now you have 51 cards. Probability of drawing second king:

Now you have 50 cards. Probability of drawing third king:

Multiply all probabilities:

4

Let us suppose that the probability of obtaining heads in a coin flip is 0.5 and the probability of the Earth being hit by an asteroid is 0.01.

The Pr(Obtaining Heads in a Coin Flip and Earth being hit by an Asteroid) .

Are these variables dependent or independent?

Independent

Dependent

Explanation

Recall that when two variables are independent,

5

Research has found that the probability of having brown eyes is and the probability of having red hair is . Assuming these probabilities are independent, what is the probability of having brown eyes and red hair?

Explanation

6

How many different combinations of 3 digit numbers can be formed using the numbers 1, 2, 3, 4, and 5, if repetitions are allowed?

125

60

120

3

15

Explanation

The key to answering this question is noting that repetitions are allowed. This means that if a number is picked, it is replaced and may be picked again, thus allowing for duplicates or triplicates. Because there are 5 choices and after each number is picked there remain 5 choices (replacement), and the question is asking for 3 digit combinations, the answer is obtained by multiplying 5 * 5 * 5 = 125. In other words, there are 5 choices for the first digit, 5 choices for the second digit, and 5 choices for the third digit.

7

If the probability of landing a heads in a coin flip is 0.5 and the probability of observing a meteor hit the earth is 0.03, and these events are independent, what is the probability of landing a heads AND observing a meteor hit the earth?

.015

.15

.035

.35

.5

Explanation

Since the two events are independent, multiply their probabilities to get their joint probability. Multiplying the probability of the coin flip, 0.5, by the probability of a meteor, 0.03, gives a probability of 0.015.

8

Automobile license plates in a certain area display three letters followed by three digits, and the letters Z and N are not used. How many plates are possible if neither repetition of letters nor of numbers is allowed?

None of the other answers

Explanation

If no letters are to be repeated and the letters Z and N are not used, then there are possible 3-letter combinations. If no digits are to be repeated, then there are possible 3-digit combinations. Multiply these two results together, and you get a total of 8,743,680 possible license plates .

9

Mike's five-person family is going out to dinner, and each person is planning to order a soda. The restaurant offers Soda1 and Soda2 only, and each family member likes both sodas equally.

What is the probability that all 5 family members will order Soda1?

Explanation

When determining the probability of several events occuring independently, you use the multiplication rule, meaning that you multiply the probability of each individual event occuring.

In this problem, the events are independent, meaning that each person's soda order does not affect the probabilty of someone else's order.

The probability of one person choosing Soda1 is 1 out of 2, or .

The probability of all five people ordering Soda1 is:

10

1 pencil and 7 pens are on a desk. One of the 8 items is randomly selected from the desk and then is replaced. Once again, an item is selected from the 8 items.

What is the probability that the pencil was selected both times?

Explanation

To find the probability of outcomes for two separate events, multiply the probability of the two outcomes. Here, the two events are independent (the second event is not affected by the result of the first) and the probability is the same for both events.

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