Questions: Bowling: Sarah and Thomas are going bowling. The probability that Sarah scores more than 175 is 0.4 , and the probability that Thomas scores more than 175 is 0.2 . Their scores are independent. Round your answers to four decimal places, if necessary. (a) Find the probability that both score more than 175. (b) Given that Thomas scores more than 175 , the probability that Sarah scores higher than Thomas is 0.2 . Find the probability that Thomas scores more than 175 and Sarah scores higher than Thomas.

Bowling: Sarah and Thomas are going bowling. The probability that Sarah scores more than 175 is 0.4 , and the probability that Thomas scores more than 175 is 0.2 . Their scores are independent. Round your answers to four decimal places, if necessary. (a) Find the probability that both score more than 175. (b) Given that Thomas scores more than 175 , the probability that Sarah scores higher than Thomas is 0.2 . Find the probability that Thomas scores more than 175 and Sarah scores higher than Thomas.
Transcript text: Bowling: Sarah and Thomas are going bowling. The probability that Sarah scores more than 175 is 0.4 , and the probability that Thomas scores more than 175 is 0.2 . Their scores are independent. Round your answers to four decimal places, if necessary. (a) Find the probability that both score more than 175. (b) Given that Thomas scores more than 175 , the probability that Sarah scores higher than Thomas is 0.2 . Find the probability that Thomas scores more than 175 and Sarah scores higher than Thomas.
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Solution

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Solution Steps

Solution Approach

(a) To find the probability that both Sarah and Thomas score more than 175, we multiply the probability of Sarah scoring more than 175 by the probability of Thomas scoring more than 175, since their scores are independent.

(b) To find the probability that Thomas scores more than 175 and Sarah scores higher than Thomas, we multiply the probability of Thomas scoring more than 175 by the conditional probability that Sarah scores higher than Thomas given that Thomas scores more than 175.

Step 1: Calculate the Probability that Both Score More than 175

To find the probability that both Sarah and Thomas score more than 175, we use the formula for the probability of independent events:

\[ P(\text{Sarah > 175 and Thomas > 175}) = P(\text{Sarah > 175}) \times P(\text{Thomas > 175}) \]

Given:

  • \( P(\text{Sarah > 175}) = 0.4 \)
  • \( P(\text{Thomas > 175}) = 0.2 \)

Substituting the values:

\[ P(\text{Sarah > 175 and Thomas > 175}) = 0.4 \times 0.2 = 0.08 \]

Step 2: Calculate the Probability that Thomas Scores More than 175 and Sarah Scores Higher than Thomas

To find the probability that Thomas scores more than 175 and Sarah scores higher than Thomas, we use the formula for conditional probability:

\[ P(\text{Thomas > 175 and Sarah > Thomas}) = P(\text{Thomas > 175}) \times P(\text{Sarah > Thomas} \mid \text{Thomas > 175}) \]

Given:

  • \( P(\text{Thomas > 175}) = 0.2 \)
  • \( P(\text{Sarah > Thomas} \mid \text{Thomas > 175}) = 0.2 \)

Substituting the values:

\[ P(\text{Thomas > 175 and Sarah > Thomas}) = 0.2 \times 0.2 = 0.04 \]

Final Answer

  • The probability that both Sarah and Thomas score more than 175 is \(\boxed{0.08}\).
  • The probability that Thomas scores more than 175 and Sarah scores higher than Thomas is \(\boxed{0.04}\).
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