Probability - Trees

The questions on this page focus on:

1. basic probability.

2. constant probability.

3. probability with replacement.

4. probability without replacement.

5. harder contexts.

6. incomplete trees.

Basic	1. The probability that a particular plant will grow to maturity is 1/6. What is the minimum number of plants which will have to be planted to have at least a 95% chance that one plant will reach maturity? Answer.(i) Pr(EE) = 5/33 (ii) Pr(neither E) = 7/22. (iii) Pr (not same language) = 47/66.
	2.A fair coin is tossed four times. Find the probability that: (i) the first three tosses are heads. (ii) there are at least three heads in the four tosses. Answer.(i) Pr(HHHH)+Pr(HHHT)=1/8 (ii) Pr (at least 3 H) = 5/16.
	3. Answer.(i) (a) Pr(GG) = 0.36 (ii) Pr(GB) = 0.48. (iii) Pr (BB) = 0.16.
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	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. There are 10 green marbles and W white marbles in a bag. The probability of selecting a whire marble is 4/9. How many more white marbles need to be added to the bag so that the probability of selecting a white marble from the bag is 3/5? Answer.7 more white marbles.
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Constant probability.	10. David and Jack are taking turns in a game with a single die. The first person to roll a 6 is the winner. David starts the game. (i) What is the probability that David wins on his first turn? (ii) By drawing a tree diagram to show the pattern of results over say the first four throws in the game, determine the probability that David eventually wins the game. Answer.(i) Pr(David wins on 1st throw) = 1/6 (ii) Pr(David wins) = 6/11.
	11. A sprinter knows that the probability of improving on a personal best time is 0.2. Find the probability that: (i) a personal best time will not be achieved in three successive races. (ii) a personal best time will be achieved at least once in three successive races. (iii) a personal best will be achieved in 2 out of 3 races. Answer.(i) 0.512. (ii) 0.458. (iii) 0.096.
	12. Alex and Erin toss a biassed coin alternately with Alex going first. The probability that the coin shows a tail on any toss is 1/3. The first person to throw a tail wins the game. What is the probability that: (i) Alex wins the game on her first throw? (ii) Erin wins the game on her first throw? (iii) after 4 tosses of the coin, there is no winner? Answer.(i) 1/3. (ii) 2/9. (iii) 16/81.
	13. The probability that a man lives to the age of 75 is 3/5 and the probability that his wife lives to the age of 7 is 2/3. By drawing a probability tree, find the probability that: (i) both the man and his wife will live to the age of 75. (ii) only the man will live to 75. (iii) only the wife will live to 75. (iv) at least one of them will live to 75. Answer.(i) 1/3. (ii) 2/9. (iii) 16/81.
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With replacement	15.
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Without replacement	20. A box contains five cards. Each card is labelled with a letter with the letters being A, B, B, C and C. A card is selected at random from the box and then a second is drawn witout replacement of the first card. (i) What is the probability that the draw resulted in B then A? (ii) What is the probability that the second card drawn had the letter C? Answer.(i) Pr(B then A) = 1/5 (ii) Pr(2nd card is C) = 2/5.
	21. In a class raffle, 40 tickets are sold. There are only two prizes. The tickets are placed in a barrel and two are drawn. Once a ticket is drawn out, it is clearly not replaced. Judy bought 3 tickets so she could support the school. Find the probability: (i) Judy wins first prize. (ii) Judy wins both prizes. (iii) Judy wins second but not the first prize. (iv) Judy does not win either prize. Answer.(i) Pr (Judy wins 1st) = 3/40. (ii) Pr (Judy wins both) = 3/40 × 2/39 = 1/260. (iii) Pr (Judy wins 2nd) = 37/40×3/39=37/520. (iv) Pr (no prize) = 37/40×36/39 = 111/130.
	22. A box contains 12 chocolates all of exactly the same appearance. Four of the chocolates are hard and eight are soft. Jim eats three chocolates chosen randomly from the box. Find the probability that: (i) the first chocolate Jim chooses is hard; (ii) Jim eats three hard chocolates. (iii) Jim eats exactly one hard chocolate. (iv) Jim has already eaten all 12 chocolates by the time you get to this part :) Answer.(i) 1/3. (ii) 1/55. (iii) 84/165. (iv) 1.000000 :).
	23. There are two teams. Team A has 5 girls and 3 boys and Team B has 9 girls and 2 boys. For the next round of competition, one person has to be seklected randomly from each team. Find the probability that: (i) two girls are selected. (ii) two boys are selected. (iii) at least one boy is selected. Answer.(i) Pr (2 girls) = 45/88 (ii) Pr (2 boys) = 3/44. (iii) Pr (≥ 1 boy) = 37/88.
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Harder	26. On any given day towards the end of the season, people on a whale watching boat off Hervey Bay in Queensland have a 40% chance of seeing at least one whale per day. (i) What is the probability of NOT seeing any whales in five days? (ii) How many days must the whale-watching boat be operating so that the chance of seeing one or more whales in that time is at least 90%? Answer.(i) Pr (no whales) = 0.077 (ii) Need 5 days for Pr (at least 1 whale) > 0.9.
Incomplete trees.	27. A school netball team has a probability of 0.8 of drawing or losing any game and a probability of 0.2 of winning any game. (i) What is the probability of the team winning at least 1 of 4 consecutive matches played? (ii) What is the least number of matches the team needs to play to be 90% sure they will have won at least one match? Answer.(i) Pr (at least 1 of 4) = 0.59. (ii) Pr (95% sure) = 10.32 so 11 games.