Question

You toss 5 coins. What is the probability that all 5 show heads?

Answer 1

Hint: To find the probability that all the five coins show head, we have to divide the number of favourable outcomes by the total number of outcomes. We know that when 5 coins are tossed, the number of times all the coins will show heads will be 1. The total number of outcomes can be found by multiplying 2 five times since there can only be 2 events for one coin.

Complete step by step solution:
We have to find the probability that all the five coins show heads. We know that when a coin is tossed, there will only be two outcomes, that is, head and tail. Here, 5 coins are tossed. So, we have to multiply 2 five times to get the total number of outcomes.
Total number of outcomes $=2\times 2\times 2\times 2\times 2={{2}^{5}}=32$
We know that probability of an event, E, is given by
$P\left( E \right)=\dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$
We know that when 5 coins are tossed, the number of times all the coins will show heads will be 1 since the other outcomes will contain tails. Therefore, the probability that all the five coins show heads can be found as follows.
$\Rightarrow P\left( \text{All head} \right)=\dfrac{1}{32}$
Hence, probability that all the 5 coins show heads is $\dfrac{1}{32}$ .

Note: To get the total number of outcomes, we have multiplied 2 five times instead of adding. This is because we are given independent events. We can also find the required probability in an alternate way.
We know that when a coin is tossed, the probability of getting a head will be $P\left( H \right)=\dfrac{1}{2}$ since the sample space is $S=\{H,T\}$ .
Thus, when 5 coins are tossed, the probability of getting heads in all the 5 coins is obtained by multiplying the probability of getting a head (when one coin is tossed) five times.
$\Rightarrow P\left( \text{All head} \right)=\dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}=\dfrac{1}{32}$
Again, we have used multiplication instead of addition because of the independent events.
We can also find the probability that all the five coins show heads by listing the sample space. This procedure will be time consuming as we have to find all the possible combinations of heads and tails.