Which of the following pairs of events is dependent?

Taking a colored ball from a basket. Without replacement, drawing another ball.

Rolling a die. Rolling the die again.

Drawing a card from a deck of 52. Replacing the card and drawing again.

Probability Of Independent And Dependent Events

1.

What does it mean for two events to be independent?
2.

Give an example of a pair of independent events.
3.

What does it mean for two events to be dependent?
4.

Probabilities can only be between 0 and 1 (both 0 and 1 included).
- A.
  
  True
- B.
  
  False
Correct Answer
A. True

Explanation
This statement is true because probabilities are always expressed as values between 0 and 1, inclusive. A probability of 0 means that an event is impossible, while a probability of 1 means that an event is certain to occur. Any value between 0 and 1 represents the likelihood of an event occurring, with values closer to 1 indicating a higher probability and values closer to 0 indicating a lower probability. Therefore, probabilities cannot exceed 1 or be negative, ensuring that they fall within the range of 0 to 1.

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5.

If the probability of events A and B are independent, the P(A and B)=

Correct Answer
P(A)xP(B)
P(B)xP(A)

Explanation
The probability of events A and B being independent means that the occurrence of one event does not affect the occurrence of the other. In such cases, the probability of both events A and B happening together can be calculated by multiplying the individual probabilities of each event. Therefore, the correct answer is P(A)xP(B), P(B)xP(A).

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6.

Which of the following pairs of events is dependent?
- A.
  
  Rolling a die. Rolling the die again.
- B.
  
  Drawing a card from a deck of 52. Replacing the card and drawing again.
- C.
  
  Taking a colored ball from a basket. Without replacement, drawing another ball.
- D.
  
  Flipping a coin three times.
Correct Answer
C. Taking a colored ball from a basket. Without replacement, drawing another ball.

Explanation
Taking a colored ball from a basket and drawing another ball without replacement is a dependent event. The probability of drawing the second ball is affected by the outcome of the first draw because there is one less ball in the basket.

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7.

If the events A and B are dependent , P(A, then B)=

Correct Answer
P(B|A)xP(A)

Explanation
The given formula, P(A, then B) = P(B|A) x P(A), represents the probability of both events A and B occurring. It states that the probability of event B happening after event A has already occurred is equal to the conditional probability of event B given event A, multiplied by the probability of event A happening. This formula is used when the occurrence of event B is dependent on event A.

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8.

A box contains 25 concert tickets. Of the box, 7 tickets are for an Usher concert, 4 tickets are to see Beyonce, 8 tickets are for All American Rejects, and the remaining 6 tickets are for a local artist concert. Student A draws a ticket and then student B draws a ticket (without student A put their ticket back). Are these events independent or dependent?

Correct Answer
Dependent

Explanation
The events of student A drawing a ticket and student B drawing a ticket are dependent. This is because the probability of student B drawing a ticket is affected by whether student A has already drawn a ticket. If student A draws a ticket for a specific concert, then there are fewer tickets available for student B to draw for that concert, making it more likely for student B to draw a ticket for a different concert. Therefore, the outcome of student A's draw affects the outcome of student B's draw, indicating that the events are dependent.

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9.

(Following from the previous question) What is the probability that Student B draws a Beyonce ticket given that Student A already drew a ticket for the Beyonce concert?

Correct Answer
12/600
1/12

Explanation
The probability that Student B draws a Beyonce ticket given that Student A already drew a ticket for the Beyonce concert can be calculated using the formula for conditional probability. The numerator of the fraction represents the number of favorable outcomes, which is 12 (since Student B can only draw one specific ticket for Beyonce). The denominator represents the total number of possible outcomes, which is 600 (since there are 600 tickets in total). Therefore, the probability is 12/600. Another way to express this probability is 1/50, which is equivalent to 1/12.

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10.

Stacy rolls a die two times. What is the probability that she will roll a 3 on the first roll and an even number on the second roll?

Correct Answer
(1/6)(1/2)
(1/6)(3/6)
1/12
3/36

Explanation
The probability of rolling a 3 on the first roll is 1/6, as there is only one outcome out of six possible outcomes. The probability of rolling an even number on the second roll is 1/2, as there are three even numbers out of six possible outcomes. Therefore, the probability of rolling a 3 on the first roll and an even number on the second roll is (1/6)(1/2), which simplifies to 1/12.

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Probability Of Independent And Dependent Events

What does it mean for two events to be independent?

Give an example of a pair of independent events.

What does it mean for two events to be dependent?

Probabilities can only be between 0 and 1 (both 0 and 1 included).

If the probability of events A and B are independent, the P(A and B)=

Which of the following pairs of events is dependent?

If the events A and B are dependent , P(A, then B)=

(Following from the previous question) What is the probability that Student B draws a Beyonce ticket given that Student A already drew a ticket for the Beyonce concert?

Stacy rolls a die two times. What is the probability that she will roll a 3 on the first roll and an even number on the second roll?