Question

If you flip a coin $10$ times what are the odds that you will get the same number of heads and tails ?

Answer 1

In the given question, we have to find the probability of the same number of heads or tails in $10$ coin throws. This is the case of binomial probability distribution. We can solve it by using the formula $P(x){ = ^n}{C_x}{p^x}{q^{n - x}}$ where getting heads can be taken as “success” and tails as “failure” or vice-versa.

Complete step by step answer:
The number of ‘Success' in a series of n experiments, where each time a yes-no question is posed, the Boolean-valued result is expressed either with success or yes or true or one (probability $p$) or failure/no/false/zero (probability $q = 1 - p$) in a binomial probability distribution.
The binomial distribution formula is for any random variable $x$, given by;
$P(x){ = ^n}{C_x}{p^x}{q^{n - x}}$
Where, $n =$ number of experiments, $p =$ Probability of Success in a single experiment and $q = 1 - p =$ Probability of Failure in a single experiment.

We can solve the sum as follows: the probability of getting same number of heads on tossing a coin-
$p = \dfrac{1}{2}$
So, we can say that-
$q = 1 - p = 1 - \dfrac{1}{2} = \dfrac{1}{2}$
Now the coin is tossed ten times so $n = 10$. Since we require the same number of heads and tails, it means that out of ten outcomes, five shall be heads and five shall be tails. Hence, we will get $x = 5$. Applying the formula, we get,
$P(x = 5) = 10{C_5}{(\dfrac{1}{2})^5}{(\dfrac{1}{2})^{10 - 5}}$
$\Rightarrow P(x = 5) = 10{C_5}{(\dfrac{1}{2})^5}{(\dfrac{1}{2})^5}$
On further simplification, then
$P(x = 5) = (\dfrac{{10 \times 9 \times 8 \times 7 \times 6}}{{5 \times 4 \times 3 \times 2 \times 1}})(\dfrac{1}{{32}})(\dfrac{1}{{32}})$
$\Rightarrow P(x = 5) = \dfrac{{63}}{{256}}$
$\therefore P(x = 5) = 0.2461$

Thus, the odds are $0.2461$ i.e. $24.61\%$.

Note: A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. For $n = 1$, i.e. a single experiment, the binomial distribution is a Bernoulli distribution. The binomial distribution formula can also be written in the form of $n$-Bernoulli trials as: $^n{C_x} = \dfrac{{n!}}{{x!(n - x)!}}$. Hence, probability can be written as: $P(x) = \dfrac{{n!}}{{x!(n - x)!}}{p^x}{q^{n - x}}$.