First slide

Theorems of probability

Question

A coin is tossed 7 times. Then the probability that at least 4 consecutive heads appear is

Moderate

A

3/16
B

5/32
C

3/14
D

1/8

Solution

Let H denote the head, T the tail and * any of the head or tail.
Then, P(H) = 1/2, P(T) = 1/2 and P() = 1. For at least four consecutive heads, we should have any of the following patterns:
Probability
(i) H H H H * * $(1 / 2)^{4} \times 1 = 1 / 16$
(ii) T H H H H * * $(1 / 2)^{5} = 1 / 32$
(iii) * T H H H H * $(1 / 2)^{5} = 1 / 32$
(iv) * * T H H H H $(1 / 2)^{5} = 1 / 32$
Since all the above cases are mutually exclusive, the probability of getting at least four consecutive heads (on adding) is 1/16 + 3/32 = 5/32.

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