Question

1.3-9. An urn contains four balls numbered 1 through 4. The balls are selected one at a time without replacement. A match occurs if the ball numbered m is the mth ball selected. Let the event $A_i$ denote a match on the $i$th draw, i = 1, 2, 3, 4. (a) Show that $P(A_i) = frac{3!}{4!}$ for each $i$. (b) Show that $P(A_i cap A_j) = frac{2!}{4!}$, $i eq j$. (c) Show that $P(A_i cap A_j cap A_k) = frac{1!}{4!}$, $i eq j$, $i eq k$, $j eq k$. (d) Show that the probability of at least one match is $P(A_1 cup A_2 cup A_3 cup A_4) = 1 - frac{1}{2!} + frac{1}{3!} - frac{1}{4!}$. (e) Extend this exercise so that there are $n$ balls in the urn. Show that the probability of at least one match is $P(A_1 cup A_2 cup ... cup A_n)$ $qquad = 1 - frac{1}{2!} + frac{1}{3!} - frac{1}{4!} + ... + frac{(-1)^{n+1}}{n!}$ $qquad = 1 - left( frac{1}{1!} - frac{1}{2!} + frac{1}{3!} - frac{1}{4!} + ... + frac{(-1)^n}{n!} ight)$. (f) What is the limit of this probability as $n$ increases without bound?

          1.3-9. An urn contains four balls numbered 1 through 4.
The balls are selected one at a time without replacement.
A match occurs if the ball numbered m is the mth ball
selected. Let the event $A_i$ denote a match on the $i$th draw,
i = 1, 2, 3, 4.
(a) Show that $P(A_i) = frac{3!}{4!}$ for each $i$.
(b) Show that $P(A_i cap A_j) = frac{2!}{4!}$, $i 
eq j$.
(c) Show that $P(A_i cap A_j cap A_k) = frac{1!}{4!}$, $i 
eq j$, $i 
eq k$, $j 
eq k$.
(d) Show that the probability of at least one match is
$P(A_1 cup A_2 cup A_3 cup A_4) = 1 - frac{1}{2!} + frac{1}{3!} - frac{1}{4!}$.
(e) Extend this exercise so that there are $n$ balls in the
urn. Show that the probability of at least one match is
$P(A_1 cup A_2 cup ... cup A_n)$
$qquad = 1 - frac{1}{2!} + frac{1}{3!} - frac{1}{4!} + ... + frac{(-1)^{n+1}}{n!}$
$qquad = 1 - left( frac{1}{1!} - frac{1}{2!} + frac{1}{3!} - frac{1}{4!} + ... + frac{(-1)^n}{n!} 
ight)$.
(f) What is the limit of this probability as $n$ increases
without bound?

$1.3-9. An urn contains four balls numbered 1 through 4. The balls are selected one at a time without replacement. A match occurs if the ball numbered m is the mth ball selected. Let the event Ai denote a match on the ith draw, i = 1, 2, 3, 4. (a) Show that P(Ai) = frac3!4! for each i. (b) Show that P(Ai cap Aj) = frac2!4!, i eq j. (c) Show that P(Ai cap Aj cap Ak) = frac1!4!, i eq j, i eq k, j eq k. (d) Show that the probability of at least one match is P(A1 cup A2 cup A3 cup A4) = 1 - frac12! + frac13! - frac14!. (e) Extend this exercise so that there are n balls in the urn. Show that the probability of at least one match is P(A1 cup A2 cup ... cup An) qquad = 1 - frac12! + frac13! - frac14! + ... + frac(-1)^n+1n! qquad = 1 - left( frac11! - frac12! + frac13! - frac14! + ... + frac(-1)^nn! ight). (f) What is the limit of this probability as n increases without bound?$

Added by Denise S.

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