Probability - Trees

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Probability - Probability trees.
Test Yourself 2 - Solutions.

Basic	1.
	2. Answer.Pr(no 6) = 0.162 (3 dp).
	3. Answer.(i) (a) Pr(GG) = 0.36 (ii) Pr(GB) = 0.48. (iii) Pr (BB) = 0.16.
	4.
	5.
	6. Answer.(i) Pr (passes one) = 0.188 (ii) Pr (passes all 3) = 0.336 (iii) Pr (at least 1) = 0.976.
	7. Answer.(i) Pr(correct) = 1/10,000 (ii) 1 - (1/10)⁴ = 3,439/10,000 (iii) 1/1,000 (iv) 1/24
	8.
	9.
Constant probability.	10. (i) What is the probability that David wins on his first turn? Prob = 1/5. (ii) determine the probability that David eventually wins the game. On 1st throw, David wins with a probability 1/6. To win on his 2nd throw, David must lose on his first throw and Jack must lose on his first throw and then David must win. So (5/6)² × 1/6. To win on his 3rd throw, David has to lose both his first 2 throws and Jack must lose both of his first 2 throws and then David must win on his third throw. So (5/6)⁴ × 1/6. So we have a series: 1/6 + (5/6)² × 1/6 + (5/6)⁴ × 1/6 + ... Sum to infinity = 6/11.




With replacement
	16.
	17.
	18.
	10.
Without replacement	11.(i) . (ii)


Harder	26. (i) Prob (not seeing a whale) = 0.6 (which is the complement). Prob (not seeing any whales in 5 days) = 0.6² = 0.077. (ii) Prob (seeing at least 1 whale in n days) has to > 0.9. But Prob (at least 1 whale) = 1 - Prob (no whales) - we need to use the complementary event!! ∴ Prob (≥ 1 whale) = 0.9 > 1 - 0.6ⁿ 0.6ⁿ > 0.1 n= log_0.60.1 - so n > 4.5 - hence 5 days.
Incomplete trees.