$\begin{array}{1 1} \large(\frac{5}{6})^4 \frac{35}{18} \\ \large(\frac{5}{6})^5 \frac{35}{18} \\ \large(\frac{5}{6})^4 \frac{70}{18} \\\large(\frac{5}{6})^5 \frac{70}{18}\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 throw at most 2 sixes in $n=6$ throws of die.

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.

Let X be the number of times we get 6 in six throws of a die. P (getting a 6) $= p = \large\frac{1}{6} \rightarrow $$q = 1 - p =\large \frac{5}{6}$

Since 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 $(q = 1 – p)$

P (throwing at least 2 sixes) $= P (X \leq 2) = P (X=0) + P (X=1) + P (X=2)$

$P (X = 0) =^{6}C_0. \large\frac{1}{6}^0.\large\frac{5}{6}^{6–0} = \large(\frac{5}{6})^6$

$P (X = 1) =^{6}C_1. \large\frac{1}{6}^1.\large\frac{5}{6}^{6–1} $ $ = 6 \times \large\frac{1}{6} \times \large(\frac{5}{6})^5 $$= \large(\frac{5}{6})^5$

$P (X = 2) =^{6}C_2. \large\frac{1}{6}^2\large\frac{5}{6}^{6–2} $$= 15 \large\frac{1}{6}^2\large(\frac{5}{6})^4$

Therefore $ P (X \leq 2) = \large(\frac{5}{6})^6$ $+ \large(\frac{5}{6})^5$ $ + 15 \large\frac{1}{6}^2\large(\frac{5}{6})^4$

$P (X \leq 2) = \large(\frac{5}{6})^4 ( (\frac{5}{6})^2 + \frac{5}{6} + \frac{15}{36})$

$P (X \leq 2) = \large(\frac{5}{6})^4 \frac{25+30+15}{36}$

$P (X \leq 2) = \large(\frac{5}{6})^4 \frac{35}{18}$

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