Practice Exam Fall 2021

#stat#stat416#review

1. A box contains 5 white, 4 red, 3 blue, and 2 green balls. Draw 4 balls without replacement. Find the probability that you get 4 different colors.

$\left(\genfrac{}{}{0ex}{}{14}{4}\right)$ total combinations. Answer: $4!\frac{5\ast 4\ast 3\ast 2}{14\ast 13\ast 12\ast 11}$

2. A box contains 3 white balls and 2 black balls. Draw a ball at random. If it’s black, put it back. If it’s white, DON’T put it back. Then draw another ball from the box.

• Find the probability that the second draw is black. $P(W_1{B}_2)+P(B_1{B}_2)=\frac{3\ast 2}{5\ast 4}+\frac{2\ast 2}{5\ast 5}=\frac{6}{20}+\frac{4}{25}=0.46$

• Find the probability that the first draw was white, given that the second draw was black. $P(W_1|B_2)=\frac{P(W_1\cap B_2)}{B_2}=\frac{0.3}{0.46}=0.652$

3. Roll a die until you get 15 aces. Find the approximate numerical probability that it takes MORE THAN 100 rolls to get 15 aces.

$\mu =100\ast 1/6=16.666$ $\sigma =\sqrt{100\ast \frac{5}{36}}=3.72$ $\frac{14.5-\mu }{\sigma }=\frac{-2.1666}{3.72}=-0.582$ $P(X<15)=\varphi (-0.582)\approx 0.281$ <- Answer

4. Roll a die 8 times.

(a). Let N be the number of different results seen among the 8 rolls. So, if the results are 66654454, the N = 3, since 4, 5, and 6 are the results seen. Find E(N).

Indicators! Let $I_k$ be indicator of seeing result ‘k’ for k = 1,2,3,4,5,6. Then N = $I_1+I_2+...+I_6$ For each $E(I_k)$ = $P\{\text{see result k}\}=1-(\frac{5}{6}{)}^8=0.7674$ $E(N)=6\ast 0.767=4.60$

(b). Find the probability that there is at least one 5 roll and at least one 6 roll. $\neg (A\wedge B)=\neg A\vee \neg B$ (P(A) + P(B) - P(A and B)) $P(r_5\ge 1)=1-P(r_5=0)=1-(\frac{5}{6}{)}^8=0.767$ $P(r_6\ge 1)=0.767$ $P(r_{5,6}\ge 1)=1-P(r_{5,6}=0)=1-(\frac{4}{6}{)}^8=0.961$ ANSWER: 0.767 + 0.767 - 0.961 = 0.573

(c). Let X be the number of 5 rolls, and let Y be the number of 6 rolls. Find E( XY ).

??? idk… ???