PROBLEM I have an MP3 player and it playes songs randomly re

PROBLEM:

I have an MP3 player and it playes songs randomly regardless of what just played. In my MP3 Player I have the following songs:

6 from The Beatles
10 from Led Zeppelin
8 from Amy Winehouse
11 from Michael Buble
15 from The Doors

I like to dance to the music from The Beatles, Led Zeppelin and Michael Buble
I like to dance to the music from The Doors and Michael Buble
I will just listen to the music from Amy Winehouse

I also have 3 kids:
Harry is embarrassed when I dance
Isaac is embarrassed when I sing
Jacklyn is embarrased half the time I sing and a third of the time I dance

a) Suppose all three kids are in the car with me, what is the probability that exactly 2 kids are embarrassed?

SOLUTION

I have calculated that the probability of them being embarrased is this:
p(Harry is embarrassed) = 27/50
p(Isaac is embarrassed) = 26/50
p(Jacklyn is embarrassed) = (1/2)(26/50) + (1/3)(27/50) = 22/50

I hope those calculations are right, but with those numbers how would I calculate that exactly two are embarrased?? Can you please show me the whole process and the correct notation, like the union of all the events or something please.

Thank you

Solution

Yes your calculations are right.
Now , the event exactly two are embarrased occurs when any one of the three is not embarresed.
So we need to see all possible combinations of two childeren = 3C2 = 3

P(Harry and Issac are embarrassed but not Jacklyn) = (27/50)*(26/50)*(1 - 22/50) = 0.157248
P(Harry and Jacklyn are embarrassed but not Issac) = (27/50)*(1 - 26/50)*(22/50) = 0.114048
P(Issac and Jacklyn are embarrassed but not Harry) = (1 - 27/50)*(26/50)*(22/50) = 0.105248

Summing above three probabilities will give probability for \" exactly two are embarrased\" :
= (0.157248+0.114048+0.105248)
= 0.376544 Answer