A six-faced fair die is thrown until 2 appear | vedclass

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(D) Let $p$ be the probability of getting a $2$ in a single throw,so $p = \frac{1}{6}$.Let $q$ be the probability of not getting a $2$,so $q = 1 - p =

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$\frac{5}{11}$

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(D) Let $p$ be the probability of getting a $2$ in a single throw,so $p = \frac{1}{6}$.Let $q$ be the probability of not getting a $2$,so $q = 1 - p = \frac{5}{6}$.The event that $2$ appears in an even number of trials means it appears on the $2^{nd}, 4^{th}, 6^{th}, \dots$ trial.This corresponds to the sequences: (not $2$,$2$) or (not $2$,not $2$,not $2$,$2$) or $\dots$The probability is given by the infinite geometric series:$P = qp + q^3p + q^5p + \dots$This is a geometric series with first term $a = qp = \frac{5}{6} 	imes \frac{1}{6} = \frac{5}{36}$ and common ratio $r = q^2 = (\frac{5}{6})^2 = \frac{25}{36}$.The sum of an infinite geometric series is $S = \frac{a}{1-r}$.$P = \frac{\frac{5}{36}}{1 - \frac{25}{36}} = \frac{\frac{5}{36}}{\frac{11}{36}} = \frac{5}{11}$.

$X = x$	$0$	$1$	$2$
$P(X = x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

$X$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X)$	$0$	$k$	$2k$	$3k$	$3k^2$	$k^2$	$2k^2$	$7k^2+k$

$A$ six-faced fair die is thrown until $2$ appears. What is the probability that $2$ appears in an even number of trials? (The die has six faces numbered $1, 2, 3, 4, 5,$ and $6$.)

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$A$ six-faced die is biased such that $3 \times P(\text{a prime number}) = 6 \times P(\text{a composite number}) = 2 \times P(1)$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice,then the mean of $X$ is.

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