there are 10 students

Question

there are 10 students : 6 females and 4 males , in professor lee's statistic class. if he chooses 4 students whast the probability that (a) choosing 3 makes (b) at last three males, and (c) less than 3 males ?

Answer 1

there are 10 students : 6 females and 4 males , in professor lee's statistic class. if he chooses 4 students whast the probability that (a) choosing 3 males (b) at last three males, and (c) less than 3 males ?

This problem uses what's called the multinomial distribution. Suppose k is the number of males selected in the chosen set of 4. The k males can be chosen in 4!/(k! * (4-k)!), or the binomial coefficient for (4,k). Then 4-k is the number of females chosen in the set of 6, and similarly, there are 6!/((4-k)!*(6-(4-k))! = 6!/((4-k)!*(2+k)!) ways to choose 4-k females from the set of 6 females. Now note that there are 10!/(4!*6!) ways to choose 4 students from the set of 10. So the total probability of choosing k males is the number of combinations satisfying the desired constraint divided by the total number of combinations:

4!/(k! * (4-k)!) * 6!/((4-k)!*(2+k)!)

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10!/(4!*6!)

Now to answer problems a-c, just put in the appropriate values of k and sum as needed. For (a) k is 3. For (b) k is 3 and 4. For (c) it's 0, 1, and 2. (Note that the answers to (b) and (c) must add to one!