Joe and Devine meet again.
J: After our last discussion about probability, I now think in odds.
D: Go on.
J: I believe my decision making has improved. I react based on probability.
D: Great. Assuming you are making decisions for your benefit, in the long run, you will be better off. Probability is the guiding force.
J: I am visiting Vegas next week. I want to use “the force” to outwit the house.
D: For that, the necessary condition is to know about conditional probability.
J: I know that probability is the long-run relative frequency. But what is conditional probability?
D: Since you are excited about Vegas, let us take the cards example.
Say, we have a deck of cards. If I shuffle and draw a card at random, what is the probability of getting a king?
J: Let me use my probability logic here. I will assume that the 52 cards will make up our sample space. Since there will be 4 kings in a deck of 52 cards, if your shuffling is fair, the likelihood of getting a king is 4/52.
D: Good. Let us call this event A → King.
What are the odds of getting a red card?
J: Since there will be 26 red cards in 52, the odds of getting a red card are 26/52.
D: Exactly. Let us call this event B → Red.
Now, If I draw a card at random, face down, and tell you that it is red, what is the probability that it will be a king?
J: So you are providing me some information about the card?
D: Yes, I am giving you a condition that the card is red.
J: Okay. Under the condition that the card is red, the odds of it being a king should be 2/26.
D: Can you elaborate.
J: Since you told me that the card is red, I only have to see how many kings are there in red cards. The sample space is now 26. There are two red kings. So the probability will be 2/26.
D: Good. Mathematically, this is written as
P(A|B) = P(A ∩ B) / P(B) or P(King | Red) = P(King and Red) / P(Red)
P(King and Red) is 2/52. P(Red) is 26/52. So we get 2/26.
J: I get it. My answer will depend on the condition. Can you provide one more example?
D: Sure. You told me that the probability of getting a king is 4/52. Suppose, this first card is faced up, and I draw another card face down. What is the probability that this second card is a king?
J: Since the first card is on the table and not replaced, the probability that the second card will be a king should be 3/51. Three kings left in the deck of 51.
D: Correct. The outcome of the second card is conditional on the outcome of the first card.
J: This makes perfect sense. If I can practice this counting and conditional probability, I can make some money on blackjack.
D: Yes you can. Knowing conditional probability is the necessary condition.
J: Why do you keep saying “necessary condition”?
D: Probability is your guiding force everywhere else in life.
But in Vegas, Joe Pesci is the force.
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