Question

Three unbiased coins are tossed. What is the probability of getting at most two heads?
(a) $\dfrac{3}{4}$
(b) $\dfrac{1}{4}$
(c) $\dfrac{3}{8}$
(d) $\dfrac{7}{8}$

Answer 1

The formula of the probability is equal to $\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}$. In the above problem, a coin is tossed thrice so the total outcomes are the thrice multiplication of 2 which could be understood as the coin is unbiased so in each toss, two possibilities are there either heads or tails. Now, favorable outcomes are the total of cases when there are 0, 1 and 2 heads will be there in the three tosses. Now, dividing favorable outcomes to the total outcomes will give us the required probability.

Complete step-by-step answer:
In the above problem, it is given that an unbiased coin has tossed three times so in each toss there are 2 possibilities can happen either head or tail can come when the coin tossed thrice then the total possibilities are the thrice multiplication of 2.
$\begin{aligned} & 2\times 2\times 2 \\\ & =8 \\\ \end{aligned}$
8 are the total possibilities that could happen in a coin tossing of three times.
Favorable possibilities are those in which at most two heads are there. It means we have to add the three cases:
Case1: when no head is there in the thrice coin tossing.
The case is only 1 when all the three coin tosses are tails.
TTT
Case2: when only one head is there in the thrice coin tossing.
The possible outcomes are:
HTT, THT, TTH
The total of the above outcomes are 3.
Case3: when only 2 heads are there in the thrice coin tossing.
The possible outcomes are:
THH, HTH, HHT
The total of the above outcomes are 3.
Now, adding the three cases we get,
$\begin{aligned} & 1+3+3 \\\ & =7 \\\ \end{aligned}$
We know that the formula of probability of any condition is equal to:
$\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}$
The favorable outcomes are 7 and the total outcomes are 8. Substituting these values in the above expression we get,
$\dfrac{7}{8}$

So, the correct answer is “Option (d)”.

Note: The mistake that could be possible in finding the favorable outcomes is that you might forgot to consider the case when there are no heads because the expression “at most 2 heads” includes the case when no heads are there in the thrice tossing of an unbiased coin so make sure you won’t miss the case when no heads are there.