Question:

Which of the following experiments have equally likely outcomes? Explain.
(i) A driver attempts to start a car. The car starts or does not start.
(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot. 
(iii) A trial is made to answer a true-false question. The answer is right or wrong.
(iv) A baby is born. It is a boy or a girl.

Updated On: Oct 31, 2023
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Solution and Explanation

(i) It is not an equally likely event, as it depends on various factors such as whether the car will start or not. And factors for both the conditions are not the same.
(ii) It is not an equally likely event, as it depends on the player's ability and there is no information given about that. 
(iii) It is an equally likely event.
(iv) It is an equally likely event.

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Notes on Probability

Concepts Used:

Conditional Probability

Conditional Probability is defined as the occurrence of any event which determines the probability of happening of the other events. Let us imagine a situation, a company allows two days’ holidays in a week apart from Sunday. If Saturday is considered as a holiday, then what would be the probability of Tuesday being considered a holiday as well? To find this out, we use the term Conditional Probability.

Let’s discuss certain theorems of Conditional Probability:

  1. Let us consider a random experiment where the sample space S is considered as space and two events namely A and B happen there. Then, the formula would be:

P(S | B) = P(B | B) = 1.

Proof of the same: P(S | B) = P(S ∩ B) ⁄ P(B) = P(B) ⁄ P(B) = 1.

[S ∩ B indicates the outcomes common in S and B equals the outcomes in B].

  1. Now let us consider any two events namely A and B happening in a sample space ‘s’, then, P(A ∩ B) = P(A).

P(B | A), P(A) >0 or, P(A ∩ B) = P(B).P(A | B), P(B) > 0.

This theorem is named as the Multiplication Theorem of Probability.

Proof of the same: As we all know that P(B | A) = P(B ∩ A) / P(A), P(A) ≠ 0.

We can also say that P(B|A) = P(A ∩ B) ⁄ P(A) (as A ∩ B = B ∩ A).

So, P(A ∩ B) = P(A). P(B | A).

Similarly, P(A ∩ B) = P(B). P(A | B).

The interesting information regarding the Multiplication Theorem is that it can further be extended to more than two events and not just limited to the two events. So, one can also use this theorem to find out the conditional probability in terms of A, B, or C.

Read More: Types of Sets

 

Sometimes students get confused between Conditional Probability and Joint Probability. It is essential to know the differences between the two.