Question

A box contains five coins. For each coin there is a different probability that a head will be obtained when the coin is tossed. (Some of the coins are not fair coins!)

Let p_i denote the probability of a head when the i th coin is tossed (i = 1, . . . , 5), and suppose that p₁ = 0, p₂ =1/4, p₃ =1/2, p₄ =3/4, p₅ =1.

The experiment we are interested in consists in selecting at random one of the coins in the box, and then tossing it a certain number of times.

Consider the events:
Ci = “The i th coin is selected”, for i = 1,2,...,5;
H_j = “The result of the j th toss is heads” (we will be interested in the cases j = 1 and j = 2);
Tj = “The result of the j th toss is tails”, (we will be interested in the cases j = 1 and j = 2);

In answering the questions below you can (and should!) use a tree diagram, but you must also write formulas for the probabilities you calculate using the events above, unions/intersections of the events above, and their probabilities/conditional probabilities. Make references to the Law of Total Probability and Baye’s Theorem when you use them.

(a) A coin is selected at random from the box (every coin has the same probability of being selected) and tossed once. What is the probability that the result of the toss is heads? Could you guess the value of this probability even without doing any calculation? Explain.

(b) A coin is selected at random from the box and when it is tossed once a head is obtained. For each i = 1, . . . , 5 find the posterior probability that the ith coin was selected. (You have to calculate five probabilities here).

(c) A coin is selected at random from the box and then tossed twice. What is the probability that the result of the second toss is heads?

(d) In the same situation as in part (c), what is the conditional probability that the result of the second toss is heads given that the result of the first toss is heads?

(e) In the same situation as in part (c), are the events “the result of the second toss is heads” and “the result of the first toss is heads” independent? Do not answer according to your intuition! You should answer after comparing the probabilities calculated in (c) and (d) and you should use the definition of independence.

(f) In the same situation as in part (c), what is the conditional probability that the result of the second toss is heads given that the result of the first toss is tails?

Answer 1

a) probability that the result of the toss is heads

=1/5 ( 0 + 1/2 + 1/4 + 3/4 +1)

= 1/2

probability of heads =1/2

note that 1 and 5 , 2 and 4 have sum of probability of getting head =1 and for coin 3 is 1/2

hence it is1/2

b)

i	p(head|i)	p(head,i)	P(i|head)
1	0	0	0
2	0.25	0.05	0.1
3	0.5	0.1	0.2
4	0.75	0.15	0.3
5	1	0.2	0.4
	2.5	0.5	1

c)

probability of result of second toss is head

= 1/2