What is chemistry..??

1.	What is chemistry..??
Answer» Answer: $\large\underline{\sf{Solution-}}$ Given that, A die is rolled 8 times and getting an even number is CONSIDERED as success denoted by event A. So, number of independent trials, n = 8. Let assume that p denotes the probability of success i.e getting an even number. and q denotes the failure i.e. not getting an even number. The possible OUTCOMES of even number when a die is rolled is { 2, 4, 6 } So, $\rm :\longmapsto\:p = \dfrac{3}{6} = \dfrac{1}{2}$ and $\rm :\longmapsto\:q = 1 - p = 1 - \dfrac{1}{2} = \dfrac{1}{2}$ Now, We know In Binomial Distribution, In a random experiment, consisting of n number of independent trials, the probability of getting r successes is $\red{\rm :\longmapsto\:\boxed{ \tt{ \: P(r) \: = \: ^nC_r \: {p}^{r} \: {q}^{n - r} \: }}}$ a) The probability that event A is success 5 times. $\red{\rm :\longmapsto\:P(5)}$ $\rm \: = \: ^8C_5 \: {\bigg[\dfrac{1}{2} \bigg]}^{5} {\bigg[\dfrac{1}{2} \bigg]}^{3}$ $\rm \: = \: \dfrac{8 \times 7 \times 6}{3 \times 2 \times 1} \times \dfrac{1}{32} \times \dfrac{1}{8}$ $\rm \: = \: \dfrac{7}{32}$ b) The probability of not getting A at all. $\red{\rm :\longmapsto\:P(0)}$ $\rm \: = \: ^8C_0 \: {\bigg[\dfrac{1}{2} \bigg]}^{8} {\bigg[\dfrac{1}{2} \bigg]}^{0}$ $\rm \: = \: \dfrac{1}{256}$ c) The probability of getting A at least one time. $\red{\rm :\longmapsto\: \displaystyle \sum^8_{r = 1} \: P(r)}$ $\rm \: = \: P(1) + P(2) + P(3) + - - + P(8)$ $\rm \: = \: 1 - P(0)$ $\rm \: = \: 1 - \dfrac{1}{256}$ $\rm \: = \: \dfrac{256 - 1}{256}$ $\rm \: = \: \dfrac{255}{256}$ ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Additional Information :- 1. Mean of Binomial Distribution = np 2. VARIANCE of Binomial Distribution = npq 3. Mean > Variance

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$\large\underline{\sf{Solution-}}$