Question

We have to randomly select 2 students for an award from a group of 5 equally deserving students. Of the five students two are female and three are male.Event C as selecting at least one female. Define event D as awarding only one male, what is the probability of event D? P(C U D) 8. P(C ∩ D) Are C and D disjoint?

Explain in detail using sample space if possible. And pls if you use symbols like this: a) D = {one male and one female} P(D) = 2C1 * 3C1 / 5C2 = 2 * 3 / 10 = 0.6 please explain what C is? what does 2C1 ,means?

Answer 1

From a group of 5 equally deserving students, 2 students can be chosen in 5!/(2!*3!)=10 ways.

Now let us define event C as selecting at least one female and event D as awarding only one male.

Now from 3 males, only one male can be chosen in ³C₁=3 ways and here for each choice the other female can be chosen in  ²C₁=2 ways.[ ⁿC_r stands for, out of n students the number ways one can draw r students.It is often called "n choose r". ⁿC_r=n!/(r!*(n-r)!].

Therefore, the probability of event D is=(3*2)/10=3/5.

Now P(C U D)=P{selecting at least one female or only one male}=P(C)+P(D)-P(C∩D).

Now, P(C)=P{selecting at least one female}=P{selecting one female or selecting two females}=P{selecting one female}+P{selecting two females}[Since they are mutually exclusive]=(²C₁*³C₁+²C₂*³C₀)/10=(6+1)/10=7/5.

P(C∩D)=P{selecting at least one female and only one male}=P{selecting one male and one female}=(²C₁*³C₁)/10=6/10=3/5.

Therefore, P(C U D)=7/5+3/5-3/5=7/5.

Now, P(C)*P(D)=21/25P(C∩D)0. Therefore, events C and D are not disjoint events.