Here are some hints/solutions for #1 and #4 from HW9-RandomVariables:
#1:
This exercise is really asking about permutations of the 10 students: imagine that the probability experiment consists of ordering the 10 students. The sample size is the 10! possible permutations. We go through each of the probabilities asked for:
- P(X=1): What is the probability of choosing a woman first? Since there are 4 women among the 10 students, the probability of choosing a woman first is 4 out of 10, i.e., P(X=1) = 4/10.
- P(X=4): For X=4, this means the first woman chosen is in the 4th position, i.e., men were chosen 1st, 2nd and 3rd, and then a woman is chosen 4th. The probability of choosing man 1st is 6/10; the subsequent (or conditional!) probability of then choosing a man 2nd is 5/9 (5 men remain among the remaining 9), and then choosing a man 3rd is 4/8 (4 men remain among the remaining 8). And then the probability of choosing one of the 4 women 4th (from the remaining 7 students) is 4/7. Thus:
P(X=4) = (6/10)*(5/9)*(4/8)*(4/7) = (6*5*4*4)/(10*9*8*7)
- P(X=6): This means 5 men have been chosen for the first five positions, and then a woman is chosen 6th. Similar to above:
P(X=6) = (6/10)*(5/9)*(4/8)*(3/7)*(2/6)*(4/5) = (6*5*4*3*2*4)/(10*9*8*7*6*5)
- P(X=9): For this we don’t need to do any calculations! Since there are only 6 men in class, it’s impossible that the highest ranked woman would be 9th. Even if we chose the 6 men for the first 6 positions, then we must choose a woman for 7th; i.e., 7 is the maximum possible value for X. Thus, P(X=9)=0.
#4:
One statement of this exercises reads:
“Let X represent the difference between the number of heads and the number of tails when a coin is tossed 46 times. Then P(X=12) = ?”
Note that this involves a binomial experiment with n=46 and p=0.5. But now we are interested in the random variable X = “the difference between the number of heads and the number of tails.”
When is X=12? This occurs in two separate outcomes of the binomial experiment:
- when H=29 and T=17
- when H=17 and T=29
We can calculate each of these binomial probabilities, either by using the binomial probability formula, or by using the spreadsheet command =binomdist(i, n, p, false). Thus, we could get P(X=12) using the spreadsheet command
=binomdist(29, 46, 0.5, false) + binomdist(17, 46, 0.5, false)
