Math / Data & Statistics

QuestionWhich of the following pairs of events are independent? Select all correct answers.
Select all that apply:
You roll a die twice. Event $A$ is getting an even number on the first roll. Event $B$ is getting a 4 on the second roll.
You roll a die twice. Event $A$ is getting a 6 on the first roll. Event $B$ is getting a total of more than 7 .
You flip a coin and roll a die. Event $A$ is getting heads on the coin. Event $B$ is getting a 3 or more on the die.
You roll a die and flip a coin. Event $A$ is getting heads with the coin and getting 5 on the die. Event $B$ is getting 3 or more on the die.

Studdy Solution

STEP 1

1. Two events are independent if the occurrence of one does not affect the probability of the other.
2. The probability of rolling any specific number on a fair die is $\frac{1}{6}$ .
3. The probability of flipping heads on a fair coin is $\frac{1}{2}$ .

STEP 2

1. Analyze the independence of events for each pair.
2. Determine if the probability of one event affects the probability of the other.

STEP 3

Consider the first pair of events:
- Event $A$ : Getting an even number on the first roll. - Event $B$ : Getting a 4 on the second roll.
Since the outcome of the first roll does not affect the outcome of the second roll, these events are independent.

STEP 4

Consider the second pair of events:
- Event $A$ : Getting a 6 on the first roll. - Event $B$ : Getting a total of more than 7.
The outcome of the first roll affects the possibility of achieving a total greater than 7, as it limits the possible outcomes on the second roll. Therefore, these events are not independent.

STEP 5

Consider the third pair of events:
- Event $A$ : Getting heads on the coin. - Event $B$ : Getting a 3 or more on the die.
The outcome of the coin flip does not affect the outcome of the die roll. Therefore, these events are independent.

STEP 6

Consider the fourth pair of events:
- Event $A$ : Getting heads with the coin and getting 5 on the die. - Event $B$ : Getting 3 or more on the die.
The occurrence of Event $A$ (getting a 5 on the die) affects the probability of Event $B$ (getting 3 or more on the die). Therefore, these events are not independent.
The independent pairs of events are:
1. You roll a die twice. Event $A$ is getting an even number on the first roll. Event $B$ is getting a 4 on the second roll.
2. You flip a coin and roll a die. Event $A$ is getting heads on the coin. Event $B$ is getting a 3 or more on the die.

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Studdy Solution

STEP 1

1. Two events are independent if the occurrence of one does not affect the probability of the other.
2. The probability of rolling any specific number on a fair die is $\frac{1}{6}$ .
3. The probability of flipping heads on a fair coin is $\frac{1}{2}$ .

STEP 2

1. Analyze the independence of events for each pair.
2. Determine if the probability of one event affects the probability of the other.

STEP 3

Consider the first pair of events:
- Event $A$ : Getting an even number on the first roll. - Event $B$ : Getting a 4 on the second roll.
Since the outcome of the first roll does not affect the outcome of the second roll, these events are independent.

STEP 4

STEP 5

Consider the third pair of events:
- Event $A$ : Getting heads on the coin. - Event $B$ : Getting a 3 or more on the die.
The outcome of the coin flip does not affect the outcome of the die roll. Therefore, these events are independent.

STEP 6

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Studdy Solution

STEP 1

1. Two events are independent if the occurrence of one does not affect the probability of the other.2. The probability of rolling any specific number on a fair die is 16 \frac{1}{6} 61​.3. The probability of flipping heads on a fair coin is 12 \frac{1}{2} 21​.

STEP 2

1. Analyze the independence of events for each pair.2. Determine if the probability of one event affects the probability of the other.

STEP 3

Consider the first pair of events:- Event A A A: Getting an even number on the first roll. - Event B B B: Getting a 4 on the second roll.Since the outcome of the first roll does not affect the outcome of the second roll, these events are independent.

STEP 4

STEP 5

Consider the third pair of events:- Event A A A: Getting heads on the coin. - Event B B B: Getting a 3 or more on the die.The outcome of the coin flip does not affect the outcome of the die roll. Therefore, these events are independent.

STEP 6

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1. Two events are independent if the occurrence of one does not affect the probability of the other.
2. The probability of rolling any specific number on a fair die is $\frac{1}{6}$ .
3. The probability of flipping heads on a fair coin is $\frac{1}{2}$ .

1. Analyze the independence of events for each pair.
2. Determine if the probability of one event affects the probability of the other.

Consider the first pair of events:
- Event $A$ : Getting an even number on the first roll. - Event $B$ : Getting a 4 on the second roll.
Since the outcome of the first roll does not affect the outcome of the second roll, these events are independent.

Consider the third pair of events:
- Event $A$ : Getting heads on the coin. - Event $B$ : Getting a 3 or more on the die.
The outcome of the coin flip does not affect the outcome of the die roll. Therefore, these events are independent.