Question

A bias coin has the probability 2/3 of turning up heads. The coin is thrown 4 times.
(a) What is the probability that the total number of heads shown is 3?
(b) Suppose that we know that outcome of the first throw is a head. Find the probability
that the total number of heads shown is 3.
(c) If we know that the total number of heads shown is 3, find the probability that the outcome
of the first throw was heads.

Answer 1

The sample space for tossing a coin 4 times contains 2⁴ = 16 outcomes. If we denote H for heads and T for tails then the sample space is S ={HHHH, HHHT, HHTT, HTTT, HTHH, HTTH, HTHT, HHTH, THHH, THHT, THTT, TTHT, TTTH, THTH, TTHH, TTTT}

Now probability for any event is defined as (number of outcomes favourable to the event/ total number of outcomes in the sample space)

A. Here our event is total number of head shown is 3. So the total number of outcomes where total number of heads is 3 is 4.

Hence by using the definition of probability our required probability is 4/16= 1/4 = 0.25

B. Here we want to find the probability of total head is 3 where the first outcome is head. So the total possible outcome where first outcome is hear is 8. Among them total head is 3 and this type of case is 3

Hence probability of total head 3 where we know that the first outcome is head is 3/8 = 0.375

C. Total number of head is 3 and number of this type of case is 4.

First outcome is head where total head is 3 and there are 3 cases of this types.

So the probability is 3/4 = 0.75