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. There are 12 students in a room. Four of them speak Arabic only, three speak Spanish only and the rest speak English only. If two students are selected at random, find the probability that: (i) both speak English. (ii) neither speaks English; (iii) they do not speak the same language. Answer.(i) Pr(EE) = 5/33 (ii) Pr(neither E) = 7/22. (iii) Pr (not same language) = 47/66.
	2. A six-sided die is rolled 10 times. What is the probability that the number 6 does not appear in the 10 rolls? Answer.Pr(no 6) = 0.162 (3 dp).
	3. In a class I know well, there are 30 students. 40% are boys and 60% are girls (girls love maths more!!). I select two students at random (we all get on well) to help me hang some great maths posters I just purchased. What is the probability that: (i) both students selected are girls? (ii) one student is a girl and the other is a boy. (iii) neither student is a girl? Answer.(i) (a) Pr(GG) = 0.36 (ii) Pr(GB) = 0.48. (iii) Pr (BB) = 0.16.
	4. In one container I have 3 red balls and 4 black balls. In a second container I have 2 red and 7 black balls. I draw one ball from each container. What is the probability of selecting: (i) 2 red balls? (ii) at least one red ball? (iii) no more than one red ball? Answer.(i) Pr(RR) = 2/21. (ii) Pr(≥ 1R) = 5/9. (iii) Pr (≤ 1) = 19/21.
	5. The probability that the temperature will fall below -10° on any day through winter at a particular city in the northern hemisphere is 0.85. What is the probability that the temperature in this city will fall below -10° on at least one day of the weekend? Answer.Pr(at least one day) = 1 - Pr(not on 2 days) = 0.98
	6. The probability that Harley will pass a Maths test is 0.6, a PDHPE test is 0.8 and a Geography test is 0.7. When he sits for these three tests at his Trial, what is the probability that Harley: (i) passes all three tests? (ii) passes exactly one of these tests? (iii) passes at least one of these tests? Answer.(i) Pr (passes one) = 0.336 (ii) Pr (passes all 3) = 0.188 (iii) Pr (at least 1) = 0.976.
	7. I have a friend who forgot her 4 digit PIN number for the ATM. The PIN is of course made from the integers 0 to 9. (i) What is the probability of my friend correctly guessing her PIN? (ii) What is the probability of her guessing at least one digit correctly? (iii) She now thinks that there is a 6 or a 7 as either the second or the last digit. What is now her probabiity of correctly guessing her PIN? (iv) Now, with a sigh of relief, my friend remembers the four digits in her PIN but not their order. What is now her chance of guessing her PIN correctly? Answer.(i) Pr(correct) = 1/10,000 (ii) 1 - (1/10)⁴ = 3,439/10,000 (iii) 1/1,000 (iv) 1/24
	8. Two BMX cross-country riders compete in a difficult event. Organisers anticipate that Jeff has a 60% chance of finishing the event while they give Ben a 40% chance of finishing. What is the probability that: (i) both riders finish the course? (ii) only one complete the course. Answer.(i) Pr(both finish) = 0.24 (ii) Prob (only 1 finishes) = 0.52.
	9.
Constant probability.	10. A biassed coin is such that the probability of getting a head is 5/8. The coin is tossed twice. (a) What is the probability of the result being: (i) two heads? (ii) two tails? (iii) a head and a tail? (b) If the first toss results in a head, what is the probability that the second toss will result in a tail? Answer.(i) (a) Pr(HH) = 25/64 (ii) Pr(TT) = 9/64. (iii) Pr (HT) = 15/32. (iv) Pr( T) = 3/8.
	11. The chance of rain on any day during July is given as being 1 in 10. What is the chance (to 3 decimal places)of: (i) rain on any three consecutive days? (ii) rain on all six weekend days in July? (iii) one 5 day school week having rain only on Monday, Wednesday and Friday? (iv) no rain during the 5 days of a school week? Answer.(i) (a) Pr(RRR) = 0.001 (ii) Pr(RR RR RR) = 0.000 - there is a chance but highly unlikely. (iii) Pr (RFRFR) = 0.001. (iv) Pr(FFFFF) = 0.590.
	12. Phoebe is starting a new job and will be catching the B-line bus from Mona Vale to the City. Her new employer tells her that if she is late on both of the first two days, she will be terminated. The probability of the bus being on time is 0.97. (i) What is the probability of Phoebe being late on the first day? (ii) What is the probability of Phoebe being late on both of the first two days? (iii) What is the probability of Phoebe keeping her job? (iv) What is the probability of Phoebe being late on only one of the first three days? Answer.(i) (a) Pr (late day 1) = 0.03 (ii) Pr (late both days) = 0.0009. (iii) Pr (keeping job) = 0.9991. (iv) Pr (late once) = 0.085 (3 dp).
	13. Roses are prone to attack by aphids so a new form of spray has been developed which does not affect the ladybirds (who eat a lot of aphids). Tests have shown that the spray is 80% successful in eliminating the aphids. Two rose plants are selected at random and sprayed. What is the probability that: (i) the aphids are eliminated on both plants? (ii) one plant is still infected with aphids? (iii) at least one plant is still infected with aphids? Answer.(i) (a) Pr (elim in both) = 16/25 (ii) Pr (elim in one) = 8/25. (iii) Pr (at least 1) = 9/25.
	14. The probability of James hitting a moving target with an arrow is 0.2. What is the least number of arrows James must shoot at a moving target if his probability to hit it at least once is to exceed 80%? Answer.8 times is the minimum.
With replacement	15. From an urn containing 4 red and 7 white balls, two balls are drawn in succession with replacement. Find: (i) The probability that the first ball drawn is white. (ii) The probability that the second ball drawn is white. (iii) The probability that both balls are white. (iv) The probability that at least one ball is white. Answer.(i) Pr(W) = 7/11 (ii) Pr(?W) = 7/11. (iii) Pr (WW) = 21/55. (iv) Pr(at least 1 W) = 49/55.
	16. A card is selected from a normal deck of 42 cards containing 13 cards of each of the suits clubs, spades (both black cards), hearts and diamonds (both are red cards). The suit is noted and then the card is returned to the deck. A second card is then chosen at random. (i) Construct a probability tree to show all the possible outcomes. (ii) What is the probability that both cards are hearts? (iii) What is the probability that both cards are red? (iv) What is the probability that one card is a club and the other card is a diamond? (v) What is the probability that at least one of the cards is a heart? Answer.(ii) Pr(HH) = 1/16. (iii) Pr(RR) = 1/4. (iv) Pr (CD or DC) = 1/4. (v) Pr(at least 1 H) = 1 - 9/16 = 7/16.
	17.
	18.
	10.
Without replacement	11.From an urn containing 4 red and 7 white balls, two balls are drawn in succession without replacement. Find: (i) The probability that the first ball drawn is white. (ii) The probability that the second ball drawn is white. (iii) The probability that both balls are white. (iv) The probability that at least one ball is white. Answer.(i) Pr(W) = 7/11. (ii) Pr(?W) = 7/11. (iii) Pr (WW) = 49/121. (iv) Pr( at least 1 W) = 105/121.
	12. Grace buys 4 tickets in a raffle. There is a total of 100 tickets being sold. The prizes on offer are $40 for first prixe,, $30 for second prize and $20 for third prize. Find the probability that Grace: (i) wins $70. (ii) wins $30. (iii) wins a prize. Answer.(i) Pr(wins $70) = 1/40425 (ii) Pr(wins $30) = 16/13475. (iii) Pr (wins a prize) = 941/8085. (iv) Pr( at least 1 W) = 49/55.
	13. There are six balls in an urn. They are numbered with the digits 1 to 6. The balls are drawn one at a time. What is the probability that: (i) the ball with the number 3 is drawn first? (ii) either the ball with the number 2 of the number 4 is drawn as the third ball? (iii) the product of the numbers on the first two balls is 12? Answer.(i) Pr(3 is 1st ball) = 1/6 (ii) Pr(2 or 4) = 1/3. (iii) Pr (product = 12) = 2/15.
	14. Two containers contain red and green counters. Container 1 has 6 green and 3 red counters while Container 2 has 4 green and 7 red counters. Yoko selects one of the containers and removes two counters without replacement. (i) Determine the probability that Yoko has selected 2 green counters from Container 2. (ii) Determine the probability that Yoko selected at least one green counter from Container 2. Answer.(i) Pr (2 green from # 2) = 3/55 (ii) Pr (≥ 1 green from # 2) = 17/55.
	15. Sam has 2 pairs of grey socks and 3 pairs of black socks. On Mondays, Wednesdays and Fridays, he selects one pair of socks at random to wear. Naturally, Sam only wears a pair of socks once!! (i) What is the probability that Sam wears a pair of black socks on Monday? (ii) What is the probabilty that Sam wears a pair of socks of the same colour on all three days? (iii) What is the probability that Sam does not wear a pair of socks of the same colour on consecutive days? Answer.(i) Pr (black) = 3/5 (ii) Pr (same colour) = 1/10. (iii) Pr (GBG or BGB) = 3/10.
	16. Alyssa is given a box of chocolates as a reward for passing her driving test. There are 10 chocolates in the box and, although they all look identical, three chocolates have caramel centres and the rest have mint centres. Alyssa - showing great restraint - only selects three chocolates at random. Find the probability that: (i) Alyssa east three mint chocolates. (ii) Alyssa eats exactly one caramel chocolate. Answer.(i) 7/24. (ii) 21/40.
	17. There are four eggs remaining in a carton. It is known that one of the eggs is cracked. Two eggs are selected at random. (i) What is the probability that one of first two eggs removed from the carton is cracked? (ii) What is the probability that the last egg removed from the carton will be cracked?
	18. A box contains five black marbles and five white marbles. Monica selects three marbles from the box without replacing them. What is the probability that at least one of the selected marbles is white? Answer.(i) 11/12.


Harder	Bag A contains 4 blue balls and 3 red balls. Bag B contains 2 blue balls and 3 red balls. A person selects one of the 7 balls from Bag A and puts it into Bag B. She then selects a ball from Bag B. What is the chance that the ball selected from Bag B is red? Answer.Pr (red from Bag B) = 4/7.
Mutually exclusive events.	21. In a game of football, the probability that Damien will tackle an opposing player is 0.7. If Damien does not tackle that player, the probability than Nathan makes the tackle on the attacking player is 0.4. Find the probability that an attacking player is tackled. Answer.Pr(tackle) = 0.82
	22. The probability that Georgia meets her perfect partner is 5%. She keeps meeting people but intends to stop when she meets her ideal person. (i) By using a tree diagram or otherwise, find the probability that Georgia will meet her perfect match when she meets the third person. (ii) Develop a expression which decribes the probability that Georgia meets her perfect partner in three or less encounters. (iii) By extending that expression, find the probability that Georgia meets her perfect match in 10 encounters. (iv) What is the smallest number of encounters Georgia needs to arrange if she is to have a probability of 95% of meeting her perfect match. Answer.(i) (a) Pr (3rd) = (0.95)²×0.05 (ii) Pr (3 or less) = 0.05 + 0.95×0.05 + 0.95²×0.05. (iii) Pr (no more than 10) = 0.401. (iv) number to meet is 59 - a great social life ahead!!.
	23. The local model boat club has eight equally boats. Three are green, four are blue and one is red. If two races are held, what is the probability that: (i) both winners are blue. (ii) neither winner is green. (iii) the winners are different colours. (iv) both winners are different colours given the amended rule that the winner of the first race is not allowed to compete in the second race. Answer.(i) (a) Pr (BB) = 0.25 (ii) Pr (not G) = 25/64. (iii) Pr (diff cols) = 19/32. (iv) Pr(BB or GG) = 9/28.
	24. Charlotte and Ben play a game of dice where they take turns in throwing two unbiassed dice. The winner is the first person to throw a double. Ben throws first (he is younger than Charlotte). (i) Show that the probability that Charlotte wins the game on his first throw is . (ii) Show that the probability Charlotte wins the game on her first or second turn is . (iii) Calculate the probability that Charlotte wins the game. Answer.(i) (a) Pr (Charlotte wins) = 6/11.