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Three groups of children contain respectively 3 girls and 1 boy, 2 girls and 2 boys and 1 girl and 3 boys. One child is selected at random from each group. What is the chance that three selected - Mathematics and Statistics

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Sum

Three groups of children contain respectively 3 girls and 1 boy, 2 girls and 2 boys and 1 girl and 3 boys. One child is selected at random from each group. What is the chance that three selected consists of 1 girl and 2 boys?

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Solution

Group-I Group-II Group-III
Girls Boys Girls Boys Girls Boys
3 1 2 2 1 3

Let G1, G2, G3 denote events for selecting a girl and B1, B2, B3 denote events for selecting a boy from 1st, 2nd and 3rd groups respectively.

Then P(G1) = `3/4`, P(G2) = `2/4`, P(G3) = `1/4`

P(B1) = `1/4`, P(B2) = `2/4`, P(B3) = `3/4`

Where G1, G2, G3, B1, B2, and B3 are mutually exclusive events.
Let E be the event that 1 girl and 2 boys are selected
∴ E = (G1 ∩ B2 ∩ B3) ∪ (B1 ∩ G2 ∩ B3) ∪ (B1 ∩ B2 ∩ G3)

∴ P(E) = P(G1 ∩ B2 ∩ B3) + P(B1 ∩ G2 ∩ B3) + P(B1 ∩ B2 ∩ G3)

= P(G1)·P(B2)·P(B3) + P(B1)·P(G2)· P(B3) + P(B1)· P(B2)·P(G3)

`=(3/4 xx 2/4 xx 3/4) + (1/4 xx 2/4 xx 3/4) + (1/4 xx 2/4 xx 1/4)`

= `(18 + 6 + 2)/64`

= `26/64`

= `13/32`

Concept: Addition Theorem of Probability
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board
Chapter 7 Probability
Miscellaneous Exercise 7 | Q 2 | Page 110
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