Question

Repeat Example $1C$when the balls are selected with replacement.

Short answer

$P(X=i)={\left(\frac{i}{20}\right)}^4$

$P(X>10)=\frac{134667}{160000}$

Step-by-step solution

Step1: Given Information

$x$ is the highest numbered ball among the four randomly selected balls, for each pick there are$20$options.

Step2: Explanation

Four balls are randomly chosen with a substitute from a pot that includes$20$balls numbered from localid="1646498376406" $1$to localid="1646498381255" $20$. X is the largest numbered ball selected that

takes on the values $1,2,3,4,5,.....20.$Want to find the probability of X in each case. Since the selection is done with a substitute, it is possible to get four balls each numbered with an identical number. Hence, X can take any of the $20$probable values.

In other terms, the question is to find the probability that X is the highest numbered ball among the four randomly selected balls. Since, the choosing is done with replacement, for each pick of balls, there are $20$choices available. Hence, the whole probable number of selections is

$S={20}^4$

$x=i$has got the highest number of selection and is given by:

$Q=i^4$

For each selection there are $i$balls and are numbered$i$or less.

Hence the probability that $X$takes on each possible values is:

$P(X=i)=\frac{i^4}{{20}^4}$

$P(X=i)={\left(\frac{i}{20}\right)}^4$

Now, P(X>10) is given by:

$P(X>10)=1-P(X\le 10)$

$P(X>10)=1-\sum _{i=1}^{10}{\left(\frac{i}{20}\right)}^4$

$P(X>10)=1-\frac{25333}{160000}$

$P(X>10)=\frac{134667}{160000}$

$P(X>10)=\frac{134667}{160000}$

Step3: Final Result

$P(X=i)={\left(\frac{i}{20}\right)}^4$

$P(X>10)=\frac{134667}{160000}$