Question

using the multiplication rule for independent events
after playing the \spin to win\ game, which has a probability of 0.25 of winning on any individual spin, nick decides to play \knock down the bottles.\ to win, nick has to knock down all 3 bottles. nick determines that the probability of winning this game is 0.8. nick plays \spin to win\ twice and then plays \knock down the bottles\ twice. what is the probability that nick loses both times he plays \spin to win\ and wins both times he plays \knock down the bottles\?
○ 0.04
○ 0.0025
○ 0.0225
○ 0.36

Explanation:

Step1: Find probability of losing "Spin to Win"

The probability of winning "Spin to Win" is \( 0.25 \), so the probability of losing is \( 1 - 0.25 = 0.75 \).

Step2: Probability of losing "Spin to Win" twice

Since the plays are independent, use the multiplication rule. The probability of losing twice is \( 0.75 \times 0.75 = 0.5625 \)? Wait, no, wait. Wait, the question is: "loses both times he plays 'Spin to Win' and wins both times he plays 'Knock down the Bottles'". Oh, I misread. Let's correct.

First, probability of losing "Spin to Win" each time: \( P(\text{lose Spin}) = 1 - 0.25 = 0.75 \). He plays it twice, so losing twice: \( 0.75 \times 0.75 \).

Probability of winning "Knock down the Bottles" each time: \( 0.8 \). He plays it twice, so winning twice: \( 0.8 \times 0.8 \).

Now, these are independent events (Spin and Knock are different games), so multiply all together: \( (0.75 \times 0.75) \times (0.8 \times 0.8) \).

Calculate \( 0.75 \times 0.75 = 0.5625 \), \( 0.8 \times 0.8 = 0.64 \). Then \( 0.5625 \times 0.64 = 0.36 \). Wait, but let's check again.

Wait, the problem says: "loses both times he plays 'Spin to Win' and wins both times he plays 'Knock down the Bottles'".

So:

• Probability of losing one "Spin to Win": \( 1 - 0.25 = 0.75 \). Two losses: \( 0.75^2 \).

• Probability of winning one "Knock down": \( 0.8 \). Two wins: \( 0.8^2 \).

Since the Spin games and Knock games are independent, multiply the probabilities of the two events (losing Spin twice and winning Knock twice).

So \( (0.75)^2 \times (0.8)^2 = (0.75 \times 0.8)^2 = (0.6)^2 = 0.36 \). Ah, that's a simpler way. \( 0.75 \times 0.8 = 0.6 \), then square it: \( 0.6^2 = 0.36 \).

So the probability is \( 0.36 \).

Answer:

0.36