Question

If the sum of the mean and variance of a binomial distribution for 5 trials is 1.8, then find the value of p, where p is the probability of success in the experiment.
A) 0.18
B) 0.2
C) 0.8
D) 0.4

Answer 1

Hint: We will first of all write the formula for mean and variance in a binomial distribution and then sum those up by putting the value of n = 5 and equating the result to 1.8. We will thus get the possible values for p and we will reject one of them using the common sense and probability facts.

Complete step-by-step answer:
Let us first discuss the mean and variance formulas for a binomial distribution:
Mean is given by np, where n is the number of trials in the experiment and p is the probability of success in trial.
Mean = np …………….(1)
Variance is given by the formula np(1-p), where n is the number of trials in the experiment and p is the probability of success in trial.
Variance = np (1 – p) ………….(2)
Now, adding (1) and (2) will result in 1.8 as per the given data in question.
So, we have:- \[np + np\left( {1-p} \right) = 1.8\]
Taking np common from LHS and rewriting it as:-
\[ \Rightarrow np\left( {1 + 1-p} \right) = 1.8\]
\[ \Rightarrow np\left( {2-p} \right) = 1.8\]
Now, putting in the value of n = 5 as per question, we will get:-
\[ \Rightarrow 5p\left( {2-p} \right) = 1.8\]
Simplifying it on both sides, we will get:-
\[ \Rightarrow 5{p^2} - 10p + 1.8 = 0\]
We know that if we have an equation $a{x^2} + bx + c = 0$, then its solution is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.
Applying this on the given equation \[5{p^2} - 10p + 1.8 = 0\], we will get:-
$p = \dfrac{{10 \pm \sqrt {100 - 4 \times 5 \times 1.8} }}{{2 \times 5}}$
On simplifying this, we will get either p = 0.2 or p = 1.8. But we know that $p \leqslant 1$ always.
Therefore, p = 0.2
Hence, the answer is 0.2

Hence, the correct option is (B).

Note: The students might make the mistake of considering both the options together but always remember to take out the one root taking care of the facts of probability or the conditions given in question if applicable.
A probability of 0 means that an event is impossible.
A probability of 1 means that an event is certain.
An event with a higher probability is more likely to occur.
Probabilities are always between 0 and 1.