Practice section 4 of 8

A box contains 5 white balls,3 red balls and 2 green balls.

(i) Three balls are taken at random with replacement. Find the probability that they all have different colours.

(ii) Five balls are taken at random without replacement. Find the probability that (a) exactly 2 are white, (b) at least 2 are white, (c) exactly 2 are red, given that at least 2 are white.

Three people A,B,C roll a fair die for a prize. The first person to roll 6 wins. Each round proceeds A, B, C and repeats until someone wins. Find the probability that (i) A wins on the first throw, (ii) C wins on the first attempt, (iii) no one wins in a round, (iv) A wins.

In darts, a player has probability 0.4 of hitting treble twenty with one dart. How many throws are necessary so that the probability of hitting treble twenty at least once exceeds 0.9?

In a game, an ordinary fair die is thrown twice and the number that appears on each throw is noted. Find the probability that

(i) both numbers are ‘6’,

(ii) neither number is a ‘5’,

(iii) one number is ‘6’ and the other is ‘5’,

(iv) the sum of the numbers is 6,

(v) the first number is less than the second given that the sum of the two numbers is 6.

Events A and B satisfy $P(A)=5/12$, $P(A\mid B')=7/12$ and $P(A\cap B)=1/8$. Find

(i) $P(B)$,

(ii) $P(A\mid B)$,

(iii) $P(B\mid A)$,

(iv) $P(A\cup B)$.

State, with reasons, whether A and B are independent or mutually exclusive.

Four persons are chosen at random from ten persons consisting of four men and six women. Three of the women are sisters. Calculate the probability that the four persons chosen will

(i) consist of four women,

(ii) consist of two women and two men,

(iii) include the three sisters.