A shooter can hit the bullseye of his target $7$ out of $10$ attempts on average. In a particula

Question

A shooter can hit the bullseye of his target $7$ out of $10$ attempts on average. In a particular training, he made $35$ attempts on the target.

State two assumptions for the number of bullseyes achieved in a training to be well modelled by a binomial distribution.[2]
Explain why one of these assumptions stated in part (i) may not hold in this context.[1]

Assume now that the number of bullseyes the shooter achieved in a training follows a binomial distribution.

Given that $m$ is the most probable number of bullseyes the shooter can achieve in a training, find the value of $m$ and state its corresponding probability.[2]
Find the probability that the shooter achieves at least $25$ bullseyes in a training.[2]
The shooter attended $40$ trainings. Using a suitable approximation, find the probability that he achieved an average of at least $25$ bullseyes per training.[3]

Accepted Answer

Answers: (i) Whether the shooter achieves / hits bullseye in an attempt is independent of whether the shooter achieves / hits bullseye in any of the other attempts. The shooter has the same probability of achieving / hitting bullseye for each / every attempt made. (ii) The shooter achieving / hitting bullseye in any attempt may not be independent (iii) ${\mathrm{P}(X=25) \approx 0.145}$ (iv) ${\mathrm{P}(X \geq 25) \approx 0.510}$ (v) ${\mathrm{P}\left(\overline{X} \geq 25\right) \approx 0.122}$

2022 NYJC P2 Q8

2022 NYJC P2 Q8

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