$\begin{array}{1 1}\large \frac{11}{243}\\\large \frac{10}{243} \\\large \frac{1}{243} \\\large \frac{5}{243} \end{array} $

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- For any Binomial distribution $B (n, p),$ the probability of x success in n-Bernoulli trials, $P (X = x) = \large^{n}C_x. p^x.q^{n–x}$ where $x = 0, 1, 2,...,n$ and $(q = 1 – p)$
- Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions: (i) There should be a finite number of trials. (ii) The trials should be independent. (iii) Each trial has exactly two outcomes : success or failure. (iv) The probability of success remains the same in each trial.

The experiment is to get 4 or more correct answers by guessing given $n=5$ questions. It is a case of Bernoulli trials as it satisfies the conditions (i) finite number of trials, (ii) independent trials, (iii) there is a definite outcome and (iv) the probability of success does not change for each trial.

Given 3 possible answers, P (getting a correct answer) $ = p = \large\frac{1}{3}$ and P (not getting a correct answer) $ = q = 1 -p = \large \frac{2}{3}$

Let X be the number of correct answers. X has a bionomial distribution, the probability of x success in n-Bernoulli trials, $P (X = x) = \large^{n}C_x. p^x.q^{n–x}$ where $x = 0, 1, 2,...,n$ and $(q = 1 – p)$

We need to calculate P (getting 4 or more correct answers) = P (X = 4) + P (X = 5).

$P (X = 4) = \large^{5}C_4\large\frac{1}{3}^4$$\times$$\large\frac{2}{3}^{5-4} $$ = 5 \times \large\frac{1}{81} $$\times\large\frac{2}{3} = \frac{10}{243}$

$P (X = 5) = \large^{5}C_5\large\frac{1}{3}^5$$\times$$\large\frac{2}{3}^{5-5}$$ = \large\frac{1}{3}^5 $$= \large\frac{1}{243}$

Therefore $P (X \geq 4) = \large\frac{10+1}{243} = \frac{11}{243}$

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