Question

) 11. Suppose you have 10 gift bags and 2 contain prizes worth $50. The other gift bags have items worth $20.

(3) a. Find the probability you select two bags with one $50 prize and one $20 prize.

(3) b. Find the probability that you select two bags and both contain $50 prizes.

(3) c. Find the probability that you select three bags and two contain $50 prizes and one contains a $20 prize.

(3) d. What is the expected value if you choose one gift bag? What does this mean?

(2) e. How many bags would you likely have to pick to choose one $50 prize? Explain.

(2) f. How many bags would you have to pick to guarantee you got one $50 prize? Explain.

(2) g. What type of probability distribution is used to solve 11c?

(2) h. How many bags would you have to pick to guarantee you got one $20 prize? Explain.

Answer 1

a) Here we'll be using the hypergeometric distribution. The probability you select two bags with one $50 prize and one $20 prize:

$\small \frac{\binom21 \binom81}{\binom{10}2}=0.3556$

b) The probability that you select two bags and both contain $50 prizes:

$\small \frac{\binom22 \binom80}{\binom{10}2}=0.0222$

c) The probability that you select three bags and two contain $50 prizes and one contains a $20 prize:

$\small \frac{\binom22 \binom81}{\binom{10}3}=0.0667$

d) If X is the number of bag with a $50 prize then, X follows a Hypergeometric distribution. Thus the expected value of X is 0.2. Hence the expected value of a gift is 0.2*50+0.8*20 = $26. It means that we can expect to win $26 on picking a bag.

e) As there are only 2 bags with $50 prize among 10 bags. Now, we will find the probability of not selecting a bag with a $50 prize till nth draw. The probability is given by:

$\small \frac{8*7*6*..(8-n+1)}{10*9*8*..(10-n+1)}$. Now we will look at the value of n for which this probability goes below 0.5.

for n=1 it is 0.8, for n=2 it is 0.6222 and for n=3 it is 0.4667. Thus we should pick at least three bags.

f) To guarantee you got one $50 prize we should pick 9 bags because in the worst-case first 8 bags contain $20 bags.

g) Hypergeometric distribution.

h) To guarantee you got one $20 prize we should pick 3 bags because in the worst-case first 2 bags contain $50 bags.