Part - Statical Amalysis
Question 6 (4 marks)
A survey shows that $35 \%$ of Australian adults participate in organised sport. If five Australian adults are selected at random, what is the probability, correct to 3 significant figures, that:
(a) Exactly one participates in organised sport
2
(b) At least one participates in organised sport
(c) Only the first adult selected participates in organised sport
Answer
\(\boxed{\text{(a) } 0.360, \text{(b) } 0.836, \text{(c) } 0.0721}\)
Step 1 ：\(P(\text{participate}) = 0.35\)
Step 2 ：\(P(\text{not participate}) = 1 - P(\text{participate}) = 0.65\)
Step 3 ：(a) \(P(\text{exactly one}) = \binom{5}{1} \times 0.35^1 \times 0.65^4 = 5 \times 0.35 \times 0.65^4 \approx 0.360\)
Step 4 ：(b) \(P(\text{at least one}) = 1 - P(\text{none}) = 1 - 0.65^5 \approx 0.836\)
Step 5 ：(c) \(P(\text{only first adult}) = 0.35 \times 0.65^4 \approx 0.0721\)
Step 6 ：\(\boxed{\text{(a) } 0.360, \text{(b) } 0.836, \text{(c) } 0.0721}\)
