Explain Joint Probability and Conditional Probability in Statistics

Joint probability is the probability of two events taking place simultaneously. Simply put, it is the probability that event X occurs at the same time as event Y. The basic assumption while calculating the joint probability of two events is that the events are independent of each other.

The formula for joint probability is:

P(X በ Y) = P(X) * P(Y)

Demonstrations: Joint Probability

1. What is the probability of you getting an 8 and a red card while drawing a card from a deck?

Solution: Using the formula, P(8 በ red) = P(8) * P(red)

= (4/52) * (26/52)

=1/26

2. A bag contains three red balls, five green balls and ten black balls. What is the probability of you getting either a red ball or a green ball when you randomly draw a ball from the bag?

Solution: The probability of getting a red ball = 3/18

The probability of getting a green ball = 5/18

The probability of getting either a red ball or a green ball = 8/18 = 4/9

Conditional Probability

Conditional probability is the probability of an event, given that some other event has already occurred.

Notation P(B|A): This notation denotes the probability of event ‘B’, given that event ‘A’ has already occurred. In conditional probability, the two events are dependent.

Formula: P(B|A) = P(B በ A)/P(A)

Demonstrations: Conditional Probability

1. In a game of cards, a player wins if he draws two cards from the same unit. If the first card is a spade, what is the probability that the player will win?

Solution: If the first card is a spade, the player is left with 12 spades and 51 cards. Thus, the probability of the second card being a spade and of the player winning the game is 12/51.

2. A student has applied to a university and has a 50% chance of getting an admission. Also, as per the university guidelines, 50% of the admitted students will get hostel accommodation. What is the probability of the student getting hostel accommodation, given that he has been admitted?

Solution: P(Hostel | Admission) = P(Hostel በ Admission)/P(Admission)

P(Hostel | Admission) = 0.5 * 0.5/0.5

P(Hostel | Admission) = 0.5

3. A bag contains four red balls and five green balls. You draw a ball from it without replacing it with another one. What is the probability of you drawing a green ball in the second draw, provided that you got a red ball in the first draw?

Solution: The probability of you drawing a red ball in the first draw is 4/9, and the probability of you drawing a green ball in the second draw is 5/8. So, the probability of you drawing a green ball after drawing a red ball is (4/9) * (5/8) = 5/18.