Binomial distribution (二項分布) サンプル問題 p.150 Exercise 5.4

Binomial formula (二項分布公式)

$$
P(k \text{ successes in } n \text{ trials}) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n-k}
$$

サンプル問題 p.150 Exercise 5.4

In a certain city district, the need for money to
buy drugs is stated as the reason for 75% of all thefts.
Find the probability that among the next 5 theft cases
reported in this district

Question (問題)

(a) exactly 2 resulted from the need for money to buy
drugs;

Solution (解き方)

(a) Exactly 2 successes (drug-related thefts):

n=5 (5 thefts reported)
k=2 (exactly 2 drug-related thefts)
p=0.75 (probability of a theft being drug-related)

$$
P(2 \text{ successes}) = \binom{5}{2} \cdot 0.75^2 \cdot (1 - 0.75)^{5-2} = 10 \cdot 0.5625 \cdot 0.0625 = 0.087890625
$$

Answer (回答)

0.08789 (approximately). almost 8 %

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