Alice, Bob, and Charlotte are looking for butterflies. They look in three separate parts of a field, so that their probabilities of success do not affect each other. Alice finds 1 butterfly with probability 17%, and otherwise does not find one. Bob fines 1 butterfly with probability 25%, and otherwise does not find one. Charlotte finds 1 butterfly with probability 45%, and otherwise does not find one. Let X be the number of butterflies that they find together. What is the expected value of X?

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warylucknow warylucknow · Answer 1 · 2020-04-27T14:06:49+02:00

Answer:

The expected number of butterflies that they find together is 0.87.

Step-by-step explanation:

Let A = number of butterflies caught by Alice, B = number of butterflies caught by Bob and C = number of butterflies caught by Charlotte.

The probability function of A is:

[tex]A=\left \{ {{0.17;\ if\ A = 1} \atop {0.83;\ if\ A = 0}} \right.[/tex]

The probability function of B is:

[tex]B=\left \{ {{0.25;\ if\ B = 1} \atop {0.75;\ if\ B = 0}} \right.[/tex]

The probability function of C is:

[tex]C=\left \{ {{0.45;\ if\ C = 1} \atop {0.55;\ if\ C = 0}} \right.[/tex]

The random variable X is denoted as the number of butterflies that they find together.

Compute the expected value of X as follows:

E (X) = E (A + B + C)

= E (A) + E (B) + E (C)

[tex]=[(0.17\times 1)+(0.83\times 0)]+[(0.25\times 1)+(0.75\times 0)]\\+[(0.45\times 1)+(0.55\times 0)]\\=0.17+0.25+0.45\\=0.87[/tex]

Thus, the expected number of butterflies that they find together is 0.87.