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Say you test drive 3 different cars. Let A1 = You like car 1; A2 = You like car 2; and A3 = You like car 3. Suppose P(A1) = .5; P(A2) = .6; P(A3)=.7 and P(A1 or A2) = .8 and P(A2 and A3) = .4 and P(A1 or A2 or A3) = .9 d. Are A2 and A3 independent events? Make sure you can show your work for this in two ways.

Respuesta :

Answer:

[tex]A_2, A_3[/tex] are not independent events.

Step-by-step explanation:

We are given the following in the question:

[tex]A_1:\text{You like car 1}\\A_2:\text{You like car 2}\\A_3:\text{You like car 3}\\P(A_1) = 0.5\\P(A_2) = 0.6\\P(A_3) = 0.7\\P(A_1\cup A_2) = 0.8\\P(A_2\cap A_3) = 0.4\\P(A_1\cup A_2\cup A_3) = 0.9[/tex]

Independent events:

  • Two events A and B is said to be independent if

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

Since,

[tex]P(A_2\cap A_3) \neq P(A_2)\times P(A_3)\\0.4\neq 0.6\times 0.7 \\0.4\neq 0.42[/tex]

Thus, they are not independent events.

Now, we evaluate

[tex]P(A_2|A_3) = \dfrac{P(A_2\cap A_3)}{P(A_3)} = \dfrac{0.4}{0.7} = 0.57\\\\P(A_2|A_3) \neq P(A_2) = 0.6[/tex]

Thus, they are not independent event.

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