Independent Events and Multiple Theorems | Essay

Essay on Independent Events and Multiple Theorems!

A set of events are said to be independent if the occurrence of any one event does not affect the chance of the occurrence of any other event of the set. Thus, the two faces of two coins falling up when they tossed together are independent of each other. Likewise, the sex of a calf to be born is independent of the sex of the calf in the previous calving by the same dam.

The multiple theorem states that the probability of the occur­rence of a set of independent events simultaneously is the product of the separate probabilities of the several independent events.

If E₁, E₂….. E_n are n independent events then

P (E₁, E₂…. E_n) = P (E₁) × P (E₂) ×…… × P (E_n).

For example, if two coins are tossed simultaneously, probabi­lity that both show the face E up is P (E₁E₂) = P (E₁) × P (E₂) = 1/2 × 1/2 = 1/4.

Problem:

In a herd 16 calves were born in a month and their male to female ratio is 5:3. If three calves are drawn from these at a time find the probability (1) they are all male, (2) they are all female, (3) they are of one sex only.

Solution:

Since the sex ratio is 5: 3 for male to female we know there are 10 male and 6 females. The number of ways in which 3 calves can be drawn from 16 is ¹⁶C₃= 16×15×14 / 1×2×3 = 560.

(1) The number of ways in which 3 male calves can be selected is ¹⁰C₃ = 10×9×8/1×2×3 =120.

So the probability of getting 3 male calves in a draw is ¹⁰C₃/¹⁶C₃ = 120/560 = 3/14

(2) The number of ways in which 3 female calves are selected in a draw is ⁶C₃ = 6×5×4/1×2×3 = 20

So that probability that 3 female calves are got when three calves drawn at a time is ⁶C₃/¹⁶C₃ = 20/560 = 1/28.

(3) In a draw of 3 calves at a time, the probability of getting all of them as one sex is the probability that the three calves are either all male or three calves selected are all female. Since these are considered exclusive cases.

Probability that the three calves are of the same sex=probability that they are all male + probability that they are all female.