If P(????)=0.3P(A)=0.3, P(????)=0.4P(B)=0.4, and P(????∪????)=0.7P(A∪B)=0.7, then P(????∩????)=P(A∩B)= . (a) Are events ????A and ????B independent? (enter YES or NO) (b) Are ????A and ????B mutually exclusive? (enter YES or NO)

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warylucknow warylucknow · Answer 1 · 2020-02-28T08:31:17+01:00

Answer:

(a) No

(b) Yes

Step-by-step explanation:

It is provided that:

P (A) = 0.30

P (B) = 0.40

P (A ∪ B) = 0.70

Compute the value of P (A ∩ B) as follows:

P (A ∩ B) = P (A) + P (B) - P (A ∪ B)

[tex]=0.30+0.40-0.70\\=0[/tex]

(a)

If events X and Y are independent then,[tex]P(X\cap Y)=P(X)\times P(Y)[/tex].

Check whether A and B are independent or not as follows:

[tex]P(A\cap B) = 0\\P(A)\times P(B)=0.30\times0.40=0.12[/tex]

[tex]P(A\cap B) \neq P(A)\times P(B)[/tex]

Thus, the events A and B are not independent.

(b)

If events X and Y are mutually exclusive then,[tex]P(X\cap Y)=0[/tex].

Check whether A and B are mutually exclusive or not as follows:

[tex]P(A\cap B) = 0[/tex]

Thus, the events A and B are mutually exclusive.