Math  /  Data & Statistics

QuestionThe probability that Zeiden scores when taking a penalty is 14\frac{1}{4}. a) Copy and complete the tree diagram below to show all the possible outcomes of Zeiden taking two penalties. b) What is the probability that he does not score the first penalty but scores the second penalty? Give your answer as a fraction in its simplest form.

Studdy Solution

STEP 1

1. The probability that Zeiden scores on a penalty is 14\frac{1}{4}.
2. The probability that Zeiden does not score is 114=341 - \frac{1}{4} = \frac{3}{4}.
3. The outcomes of the two penalties are independent events.

STEP 2

1. Construct the tree diagram for two penalties.
2. Calculate the probability for the specific outcome where Zeiden does not score the first penalty but scores the second penalty.

STEP 3

Draw the first level of the tree diagram for the first penalty with two branches: "Score" and "No Score".
- Probability of "Score" on the first penalty: 14\frac{1}{4} - Probability of "No Score" on the first penalty: 34\frac{3}{4}

STEP 4

Extend the tree diagram to the second penalty for each outcome of the first penalty.
For the "Score" branch of the first penalty: - Probability of "Score" on the second penalty: 14\frac{1}{4} - Probability of "No Score" on the second penalty: 34\frac{3}{4}
For the "No Score" branch of the first penalty: - Probability of "Score" on the second penalty: 14\frac{1}{4} - Probability of "No Score" on the second penalty: 34\frac{3}{4}

STEP 5

Identify the path in the tree diagram where Zeiden does not score the first penalty but scores the second penalty.
- First penalty: "No Score" \rightarrow Probability: 34\frac{3}{4} - Second penalty: "Score" \rightarrow Probability: 14\frac{1}{4}

STEP 6

Calculate the probability for the path "No Score" on the first penalty and "Score" on the second penalty by multiplying the probabilities along the path.
Probability=(34)×(14)=316 \text{Probability} = \left(\frac{3}{4}\right) \times \left(\frac{1}{4}\right) = \frac{3}{16}
The probability that Zeiden does not score the first penalty but scores the second penalty is:
316 \boxed{\frac{3}{16}}

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