Practice section 7 of 8

Mr Ong chooses either a Beef Combo Meal or a Chicken Combo Meal at a fast food restaurant. On his first visit, the probability of beef is 0.8. Thereafter, the probability of beef is 0.6 if he chose beef on the previous visit and 0.7 otherwise.

(i) Show that the probability of beef on each of the first three consecutive visits is 0.288. [1]

(ii) Find the probability of exactly one beef meal in those visits. [2]

(iii) Given exactly two chicken meals in those visits, find the probability that the second visit’s meal was beef. [2]

A box contains 45 balls distinctly labelled 1 to 45. Seven are drawn randomly without replacement and their labels are the winning numbers. Andy randomly chooses eight distinct integers from 1 to 45.

(i) Show that the probability none of Andy’s numbers matches the winning numbers is 0.227. [1]

(ii) Hence find the probability exactly four match, given at least one match. [3]

An Art Club has 10 members:2 IP1,2 IP2,3 JC1 and 3 JC2. They take a group photo. Find the number of arrangements with all members

(i) in a row with the 2 IP1 members together and no JC2 members together, [2]

(ii) in two equal rows, one in front of the other, with the 4 IP members at the four corners. [2]

Afterwards all members sit at a round table. Find the probability that the 2 IP1 members are seated together. [2]

A bag initially contains 3 white and 6 black balls. A player draws 3 balls, one at a time, without replacement. Three white balls win the grand prize; exactly two whites win a consolation prize; otherwise the player wins nothing.

Find the probability that

(i) a player wins the grand prize,

(ii) a player wins a consolation prize,

(iii) a player wins a consolation prize, given that the first draw is black.

A and B play a racquet match of at most 3 sets. The first to win two sets wins. A wins the first set with probability 0.6. After the first set, A wins the next set with probability p given A won the preceding set; B wins the next set with probability 0.7 given B won the preceding set.

Find, in terms of p, the probability that A wins the match.

(i) The probability B won the first set given B lost the match is 0.15. Find p. [2]

(ii) Given instead p=0.65 and that A’s third win occurs in the nth match with probability 0.10866 correct to 5 significant figures, write an equation for n and solve it numerically. [3]

Express all answers as proper fractions. In 2013,60% of the unemployed citizens of country A were graduates. Within a year,70% of these unemployed graduates found a job, while 40% of the unemployed non-graduates found one.

(i) A person unemployed in 2013 is chosen at random. Find the probability the person is not a graduate, given no job has been found. [3]

(ii) Two people unemployed in 2013 are chosen at random. Find the probability that

(a) exactly one found a job within a year, [2]

(b) exactly one is a graduate, given both found jobs within a year. [3]