A fair coin is tossed four times. Calculate the probability of obtaining:
at least one head;
equal number of heads and tails.
The probabilities that Sani, Kalu and Tato will hit a target are respectively. If all the three men shoot once, what is the probability that the target will be hit only once?
This question was also reported to be attempted by majority of the candidates and
they performed well in it. Their performance in part (a) was also reported to be better than what it was in part (b).
In part (a) majority of the candidates were reported to show that the probability followed a binomial distribution nCrprqn-r, where n = 4, r = number of favourable outcome, p = probability of obtaining a head = and q = probability of not obtaining a head = .
Probability of getting at least one head = 1 – probability of getting no head
i.e. 1 – 4Co ()o ()4 = . Probability of equal number of heads and tails =
4C2 () 2()2 = x ()4 = .
In part (b), it was reported that majority of the candidates did not show that the probability that the target will be hit only once implied probability that (only Sani will hit or only Kalu will hit or only Talo will hit). The probability that Sani will not hit the target = 1 – probability that he will hit = 1 - = . In a similar way, Probability that Kalu will not hit = 1 - = while probability that Tato will not hit = 1 - = . Therefore probability that only one of them would hit =
( ´ ´ ´ ´ + ( ´ ´ ) = + + = .
respectively. If all the three men shoot once, what is the probability that the target will be hit only once?
and q = probability of not obtaining a head = 
.
)o (
)4 = 
. Probability of equal number of heads and tails =
) 2(
)2 = 
x (
)4 = 
.
= 
. In a similar way, Probability that Kalu will not hit = 1 - 
= 
while probability that Tato will not hit = 1 - 
= 
. Therefore probability that only one of them would hit = 
´ 
´ 
´ 
´ 
+ (
´ 
´ 
) = 
+ 
+ 
= 
.