NECO 2023 · Paper 1 · Q47✱

Probability & distributions

In an examination, $75\%$ of the candidates passed. Use the binomial distribution to calculate the probability that a random sample of 8 candidates contains at most 1 pass. Correct to 4 decimal places.

Worked solution (try it first)

Let $X$ be the number who pass: $X \sim B(8, 0.75)$ , with $p = 0.75$ for a pass and $q = 0.25$ for a failure.
At most 1 pass means $X = 0$ or $X = 1$ : $P(X \le 1) = q^8 + 8pq^7$ .
$0.25^8 \approx 0.0000153$ and $8(0.75)(0.25)^7 \approx 0.0003662$ .
Add them: about 0.000381.
To 4 decimal places this is 0.0004, option B.

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