Question

Two persons A and B have respectively (n+1) and n coins which they toss simultaneously. The probability that A will have more number of heads is:
$A.\dfrac{1}{4} \\\ B.\dfrac{1}{3} \\\ C.\dfrac{1}{2} \\\ D,\dfrac{1}{8} \\\$

Answer 1

Hint : The set of all possible outcomes of an experiment is said to be the sample space of the experiment.
The sample space for an experiment of tossing a coin is S={head-head, head-tail, tail-head, tail-tail}.
Any subset of the sample space is called an event.

Complete step by step solution:
Associated with each outcome in the sample space is a real number called the probability of the outcome representing the chance of happening of the event. Therefore probability of an event is
$P= \dfrac{\text{number of outcomes favourable}}{\text{total number of outcomes}}$
According to the question we have two persons one with $n+1$ and the other with $n$ number of coins.
Let the number of heads that the person A get be a and the number of tails be a’
Similarly the person B has the number of heads as b and tail as b’.
Now we know that the total outcomes (n+1)=a+a’ for A and the total outcomes for B as n=b+b’. since A has more number of outcomes therefore probability would be an inequality a>b.
The probability for A to have more heads, therefore 1-P probability of the opposite event a≤b at the same time which represents probability of A throwing more tails than B.

$$a \leqslant b \Rightarrow n + 1 - a' \leqslant n - b' \\\ \Rightarrow 1 - a' \leqslant b' \;$$

Multiplying by -1 on both the sides of inequality changing the sign of inequality and adding 1 on both the sides,

$$\Rightarrow a' \geqslant b' + 1 \\\ \Rightarrow a' > b' \;$$

Therefore we observe that both the events that A will throw more heads than B and A will throw more tails than B are equally to occur that is
$\therefore P = 1 - P \\\ 2P = 1 \\\ P = \dfrac{1}{2} \;$
So, the correct answer is “Option C”.

Note : In case of a fair coin each of the two outcomes is equally likely to occur and both the outcomes are mutually exclusive as the occurrence of one rules out the possibility of the other. Two or more events are mutually exclusive when both of them cannot occur simultaneously which is the case of tossing a coin.