Questions: In three independent flips of a coin where there is a 65% chance of flipping a head, let A denote first flip is a head, B denote second flip is a head, C denote first two flips are heads, and D denote three heads on the first three flips. Find the probabilities of A, B, C, and D, and determine which, if any, pairs of these events are independent. P(A)= (Round to two decimal places as needed.)

In three independent flips of a coin where there is a 65% chance of flipping a head, let A denote first flip is a head, B denote second flip is a head, C denote first two flips are heads, and D denote three heads on the first three flips. Find the probabilities of A, B, C, and D, and determine which, if any, pairs of these events are independent.
P(A)= (Round to two decimal places as needed.)

Transcript text: In three independent flips of a coin where there is a $65 \%$ chance of flipping a head, let A denote \{first flip is a head\}, B denote \{second flip is a head\}, C denote \{first two flips are heads\}, and D denote \{three heads on the first three flips\}. Find the probabilities of A, B, C, and D, and determine which, if any, pairs of these events are independent. $P(A)=$ $\square$ (Round to two decimal places as needed.)

Solution

Solution Steps

To solve this problem, we need to calculate the probabilities of each event based on the given probability of flipping a head. Then, we will check for independence between pairs of events by comparing the joint probability of two events with the product of their individual probabilities.

Step 1: Calculate the Probability of Each Event

We are given that the probability of flipping a head is $ p_{\text{head}} = 0.65 $.

The probability of event $ A $ (first flip is a head) is: \[ P(A) = p_{\text{head}} = 0.65 \]
The probability of event $ B $ (second flip is a head) is: \[ P(B) = p_{\text{head}} = 0.65 \]
The probability of event $ C $ (first two flips are heads) is: \[ P(C) = p_{\text{head}} \times p_{\text{head}} = 0.65 \times 0.65 = 0.4225 \]
The probability of event $ D $ (three heads on the first three flips) is: \[ P(D) = p_{\text{head}} \times p_{\text{head}} \times p_{\text{head}} = 0.65 \times 0.65 \times 0.65 = 0.2746 \]

Step 2: Check for Independence Between Pairs of Events

Two events $ X $ and $ Y $ are independent if $ P(X \cap Y) = P(X) \times P(Y) $.

Independence of $ A $ and $ B $: \[ P(A \cap B) = P(C) = 0.4225 \] \[ P(A) \times P(B) = 0.65 \times 0.65 = 0.4225 \] Since $ P(A \cap B) = P(A) \times P(B) $, events $ A $ and $ B $ are independent.
Independence of $ A $ and $ C $: \[ P(A \cap C) = P(C) = 0.4225 \] \[ P(A) \times P(C) = 0.65 \times 0.4225 = 0.2746 \] Since $ P(A \cap C) \neq P(A) \times P(C) $, events $ A $ and $ C $ are not independent.
Independence of $ A $ and $ D $: \[ P(A \cap D) = P(D) = 0.2746 \] \[ P(A) \times P(D) = 0.65 \times 0.2746 = 0.1785 \] Since $ P(A \cap D) \neq P(A) \times P(D) $, events $ A $ and $ D $ are not independent.
Independence of $ B $ and $ C $: \[ P(B \cap C) = P(C) = 0.4225 \] \[ P(B) \times P(C) = 0.65 \times 0.4225 = 0.2746 \] Since $ P(B \cap C) \neq P(B) \times P(C) $, events $ B $ and $ C $ are not independent.
Independence of $ B $ and $ D $: \[ P(B \cap D) = P(D) = 0.2746 \] \[ P(B) \times P(D) = 0.65 \times 0.2746 = 0.1785 \] Since $ P(B \cap D) \neq P(B) \times P(D) $, events $ B $ and $ D $ are not independent.
Independence of $ C $ and $ D $: \[ P(C \cap D) = P(D) = 0.2746 \] \[ P(C) \times P(D) = 0.4225 \times 0.2746 = 0.1160 \] Since $ P(C \cap D) \neq P(C) \times P(D) $, events $ C $ and $ D $ are not independent.