Question

Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes:

Outcome	3 heads	2 heads	1 head	No head
Frequency	23	72	77	28

If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up.

Answer 1

Hint: This is a simple example of probability. We are given the data for various events happening out of the sample set. From the data, we can see that all the 4 given events are mutually exclusive and independent events. We know that the probability of any event happening is given by the relation $P\left( A \right)=\dfrac{n\left( A \right)}{n\left( S \right)}$, where n(A) is the number of elements in event set A and n(S) is the number of elements in the sample event set S.

Complete step-by-step answer:
It is given to us that the coin is tossed simultaneously 200 times. The outcome of one coin is totally independent of the outcome of the other coin.
From the data given, we know that the number of times the outcome of all the coins was heads is 23. The number of times two of the three coins had heads is 72 and the number of times only one of the three coins had heads is 77. The number of times the outcome of the toss had no heads is 28.
Let S be the sample event set of all the possible outcomes of tossing of 3 coins simultaneously 200 times. The number of elements in the sample event set S will be n(S) = 200.
We are supposed to find the probability of getting the outcome as two heads.
Let A be the event set of all the times when the outcome of tossing of 3 coins out of 200 times had two heads. The number of elements in the event set A will be n(A) = 72
We know that the probability of any event happening is given by the relation $P\left( A \right)=\dfrac{n\left( A \right)}{n\left( S \right)}$, where n(A) is the number of elements in event set A and n(S) is the number of elements in the sample event set S.
$\begin{align}
  & \Rightarrow P\left( A \right)=\dfrac{72}{200} \\
 & \Rightarrow P\left( A \right)=\dfrac{9}{25} \\
\end{align}$
Therefore, the probability of getting 2 heads is given as $\dfrac{9}{25}$.

Note: We are able to apply the simple relation of probability given by $P\left( A \right)=\dfrac{n\left( A \right)}{n\left( S \right)}$ only because the given events are independent and mutually exclusive. If the events would have been dependent, it would require advanced concepts of probability.