Question

One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary outcomes S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted} You are told that the chances of John’s promotion is same as that of Gurpreet, Rita’s chances of promotion are twice as likely as Johns. Aslam’s chances are four times that of John. If A = {John promoted or Gurpreet promoted}, find P(A).

Answer 1

Let E₁, E₂, E₃ and E₄ be the events that John promoted, Rita promoted, Aslam promoted and Gurpreet promoted respectively.

∴ Sample space S = {E₁, E₂, E₃, E₄}

Given that probability of John’s promotion is same as that of Gurpreet

∴ P(E₁) = P(E₄)

Rita’s chances of promotion are twice as likely as John

∴ P(E₂) = 2P(E₁)

And Aslam’s chances of promotion are 4 times that of John

∴ P(E₃) = 4P(E₁)

Since, the sum of all the probabilities = 1

∴ P(E₁) + P(E₂) + P(E₃) + P(E₄) = 1

⇒ P(E₁) + 2P(E₁) + 4P(E₁) + P(E₁) = 1

⇒ 8P(E₁) = 1

⇒ P(E₁) = ${\displaystyle \frac{1}{8}}$

P(John promoted or Gurpreet promoted) = P(E₁ ∪ E₄)

⇒ P(E₁ ∪ E₄) = P(E₁) + P(E₄) – P(E₁ ∩ E₄)

= ${\displaystyle \frac{1}{8}+\frac{1}{8}-0}$   ......[∵ P(E₁ ∩ E₄) = 0]

= ${\displaystyle \frac{1}{4}}$