Question

The number of months during the $4 -$ month monsoon season (June to September) in which the total rainfall was greater than $5$ metres, $R$ , has been recorded at a location in Meghalaya for the past $32$ years, and is shown in the following table.

Number of months	$0$	$1$	$2$	$3$	$4$
Number of years	$2$	$8$	$12$	$8$	$2$

The distribution of $R$ is to be modelled by $R ~ B (4, p)$ . Find the value of $p$ and state clearly what this value represents. Give a reason why, in real life, it is unlikely that a binomial distribution could be used to model these data accurately.

Accepted Answer

The number of months during the $4 -$ month monsoon season (June to September) in which the total rainfall was greater than $5$ metres, $R$ , has been recorded at a location in Meghalaya for the past $32$ years, and is shown in the following table.

Number of months $0$ $1$ $2$ $3$ $4$

Number of years $2$ $8$ $12$ $8$ $2$

The distribution of $R$ is to be modelled by $R ~ B (4, p)$ .

If $X$ is a discrete random variable which is binomial distributed , $X ~ B (n, p)$ then the probability of obtaining $r$ successes out of $n$ independent trials, when $p$ is the probability of success for each trial, is $P (X = r) = (\begin{matrix} n \\ r \end{matrix}) p^{r} {(1 - p)}^{n - r}$

So, according to the question, we have: $n = 4$

Therefore, $P (no months with more than 5 m)$ i.e, $32 (\begin{array}{l} 4 \\ 0 \end{array}) \times p^{0} \times (1 - p)^{4} = 2$

$\Rightarrow 32 \times \frac{4!}{0! (4 - 0)!} \times (1 - p)^{4} = 2$ [Using $(\begin{array}{l} n \\ r \end{array}) = \frac{n!}{(n - r)! r!}$ ]

$\Rightarrow (1 - p)^{4} = \frac{1}{16}$

$\Rightarrow (1 - p) = \frac{1}{2}$

$\Rightarrow p = \frac{1}{2}$

Here, $p$ represents the probability of more than $5$ metres of rainfall in any given month of the monsoon season.

So, in real life the probability of more than $5$ metres of rainfall in any given month of the monsoon season is unlikely to be constant or Whether one month has more than 5 metres of rainfall is unlikely to be independent of whether another month has.

Important Questions on The Binomial and Geometric Distributions

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