In this experiment,a class of $50$ students performs the following experiment. Each student in the class:

$$ Shuffles a standard deck of $52$ cards $($which has $13$ cards of each of the four suits: clubs, diamonds, hearts, spades$).$

$$ Picks a card from the deck.

$$ Records whether it was a diamond or not.

$$ Puts the card back and reshuffles the deck.

$$ Does this a total of $10$ times.

$$ Records X $=$ # of times a diamond was not drawn $($over the $10$ times$).$

$($a$)$ Theoretically, X $$ Bin$(,)$

$($b$)$ The frequency table is given below. Fill in the relative frequency column:

X Frequency Relative Frequency

$00$

$11$

$20$

$30$

$40$

$52$

$67$

$715$

$813$

$98$

$104$

$($c$)$ Calculate X $($the mean of X$)$ and s $($the sample standard deviation$).$

$($d$)$ Build the theoretical probability distribution function on the distribution function you found in part $($a$).$

X P$($X$)$

$0$

$1$

$2$

$3$

$4$

$5$

$6$

$7$

$8$

$9$

$10$

$($e$)$ Calculate $\backslash$mu X and $\backslash$sigma $.$

$($f$)$ Using the theoretical probability distribution function, calculate P$(5<$ X $<9).$ Either

show the numbers you added or the function you used in Excel.

$($g$)$ Using the theoretical probability distribution function, calculate P$($X $<=3).$ Either show the numbers you added or the function you used in Excel.

$($h$)$ Suppose that the experiment had been repeated $500$ times. Would you expect either of the two tables to change? If so$,$ which one$($s$)$ and how would it $($they$)$ change? Why? Why wouldn$$t the other table$($s$)$ change?