A committee of 4 representatives is to be selected from 6 Democrats and 4 Republ

Question

A committee of 4 representatives is to be selected from 6 Democrats and 4 Republicans. Find the probability that (show work and express answer as reduced fraction) a. all are democrats b. two are democrats and two are republicans c. all are republicans d. at least two are republicans If on a given day in December the probability of rain is 30% and the probability of snow is 60%. What is the probability of either ran or snow (assuming for this problem that it can't rain or snow on the same day)? A weather forecaster says that the probability it will rain on Saturday or Sunday is 50%, the probability it will rain on Saturday is 20%, and the probability it will rain on Sunday is 40%. What is the probability that it will rain both Saturday and Sunday.

Explanation / Answer

Q.2 Here there are total 10 number of people, out of which 4 have to be selected where 6 are democrats and 4 are republicans.

so, Total Number of possibilities = 10C4

(a) all are democrates.

Total Possibilites = 6C4

so Pr( all are democrates) = 6C4 /10C4 = 15/210 = 1/14

(b) Two are democrats and two are republicans

Total Possibilites = 6C24C2 =

so Pr( all are democrates) = 6C24C2/10C4 = 3/7

(C) all are republicans

Total Possibilites =  4C4 = 1

so Pr( all are democrates) =4C4/10C4 = 1/210

(D) at least two are repulblicans

so total possibilites = 10C4 - no republicans - one repulblican

= 10C4 -6C4 - 6C34C1 = 210 - 15 - 80 = 115

so atleast two are republican = 115/210 = 23/ 42

Q.3 P(Rain) =0.3

P (snow)= 0.6

P ( either rain or either snow) = P(rain) P(no snow) + P(no rain) P(snow)

= 0.3 * 0.4 + 0.7 *0.6 = 0.54

Q.4 P (saturday or sunday) = 0.5

P(saturday) = 0.2

P(sunday) = 0.4

P(only saturday) = 0.2 - P(saturday and sunday both)

P(only sunday) = 0.4 - P(saturday and sunday both)

P(only saturday) + P(only sunday) = (0.2 - P(saturday and sunday both)) + (0.4 - P(saturday and sunday both))

P(saturday or sunday) = 0.6 - 2 * P(saturday and sunday both)

so 2 * P(saturday and sunday both) = .0.6- P(saturday or sunday) = 0.6 -0.5 = 0.1

P(saturday and sunday both) = 0.05

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