### All AP Statistics Resources

## Example Questions

### Example Question #1 : How To Identify Independent Events

Given a pair of fair dice, what is the probability of rolling a 7 in one throw?

**Possible Answers:**

1/6

1/3

5/6

1/2

1/4

**Correct answer:**

1/6

There are 36 total outcomes for this experiment and there are six ways to roll a 7 with two dice: 1,6; 6,1; 2,5; 5,2; 3,4; and 4,3. Thus, 6/36 = 1/6.

### Example Question #1 : Independent And Dependent Events

A fair coin is tossed into the air a total of ten times and the result, heads or tails, is the face landing up. What is the total number of possible outcomes for this experiment?

**Possible Answers:**

512

100

2

20

1024

**Correct answer:**

1024

There are two outcomes in each trial of this experiment, and there are ten total trials. Thus, 2 raised to the tenth power yields an answer of 1024.

### Example Question #1 : How To Identify Independent Events

Mary randomly selects the king of hearts from a deck of cards. She then replaces the card and again selects one card from the deck. The selection of the second card is a(n)_________ event.

**Possible Answers:**

independent

dependent

**Correct answer:**

independent

The selection of the second card is an independent event because it is unaffected by the first event. If the king of hearts had not been replaced, then the probability of selecting a particular card would have been affected by the first event, and the second selection would have been dependent. This, however, is not the case in this question.

### Example Question #1 : Independent And Dependent Events

What is the probability of getting a sum of when rolling two six-sided fair dice?

**Possible Answers:**

**Correct answer:**

The sample space, or total possible outcomes, when rolling two six-sided dice is .

Ways to get what you want:

So there are ways to get a .

So the probability becomes

### Example Question #1 : How To Identify Independent Events

Each answer choice describes two events. Which of the following describes independent events?

**Possible Answers:**

A card is drawn from a pile and set aside. A second card is drawn.

A marble is randomly drawn from a bag and set aside. A second marble is drawn from the bag.

A shirt is randomly selected from a drawer and is placed in the laundry. A second shirt is randomly slected from the drawer.

One of ten differently colored candies is selected from a jar and eaten. A second candy is then randomly selected.

A coin is flipped in the air and the result is heads. The coin is flipped in the air again.

**Correct answer:**

A coin is flipped in the air and the result is heads. The coin is flipped in the air again.

Two events are independent of each other when the result of one does not affect the result of the other. In the case of the coin being flipped, the first result in no way influenced the result of the second coin flip. In contrast, when a card is removed from a deck of cards and set aside, that card cannot be selected when a second card is taken from the deck.

### Example Question #1 : Independent And Dependent Events

Events and are known to be independent.

while . What must be?

**Possible Answers:**

**Correct answer:**

Because the two events are known to be independent, then the following is true by definition.

.

This then becomes an algebra problem:

### Example Question #6 : Independent And Dependent Events

Two events and are independent, and while . What is ?

**Possible Answers:**

**Correct answer:**

Because the two are independent, the calculation becomes the product of the two by definition.

We need to recall that respresents the compliment of A which is everything that is not in A or in mathematical terms:

.

Likewise for the compliment of B:

Therefore to find the intersection of these two independent events we multiply them together.

### Example Question #7 : Independent And Dependent Events

True or false:

A family has 3 boys. The probability that the fourth child will also be a boy is less than 50%

**Possible Answers:**

Sometimes

More information is needed.

True

False

**Correct answer:**

False

The gender of each child can be considered an independent event. Each child has a 50% chance of being a boy, and whether a boy was already born previously does not affect the next child's gender.

### Example Question #8 : Independent And Dependent Events

True or false: When drawing two cards with replacement, the event drawing a spade first is independent of the event drawing a heart second.

**Possible Answers:**

More information is needed.

False

Sometimes

True

**Correct answer:**

True

These two events are independent of one another. During sampling with replacement, the first card does not affect the second card being picked.

To illustrate, consider the probability of drawing a heart first

Assuming you first drew a heart and replaced it in the deck, does the probability of drawing a heart as the second card change?

The probability remains the same, there are still 13 hearts and 52 total cards.

### Example Question #1 : Independent And Dependent Events

True or False: When 2 cards are drawn without replacement from a regular deck of 52 cards, the event of drawing a heart first independent of the event of drawing a heart second.

**Possible Answers:**

True

Sometimes

False

Cannot be determined by the information given.

**Correct answer:**

False

These events are not independent, because if one event happens, it affects the probability of the other event happening. Consider the probability of drawing a heart and the probability of getting a heart given a heart was already drawn. If these two probabilities are the same, the events are independent. If the two probabilities are not the asme, the events are not independent.

After a heart has already been drawn, there are now only 52 cards total and 12 hearts left. These two probabilities are not equal, therefore the events are not independent.

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