Lesson 10 – The fight for independence

I don’t have a “get out of jail free” card.

I don’t want to pay $50 to the bank because I am short on cash.

๐Ÿ˜‰ I am trusting my magic dice to fight for my freedom. I know I will roll a double.

๐Ÿ™ I am disappointed. As I wait for my turn, I realized that I did not kiss the dice before rolling. So I do it now and roll again.

๐Ÿ˜ฏ Maybe I should have kissed the dice two times since it is the second try. Oh, I did not pray before rolling. So I pray and roll the dice with optimism.

๐Ÿ˜ก I don’t believe this. My magic dice betrayed me. I will throw them away and get new ones.

Wait. The magic dice did not betray you. It is just following the probability rule for independent events. Unlike you, your magic dice does not have a memory. It does not know that the previous try was not a double. All it knows is that the probability of getting a double on any try is 16.66%.

Assume A is the event of seeing a double, and B is a previous event, say {6,1} – not a double.

The probability of getting a double given that the last try was not a double, P(A|B) is equal to the probability of getting a double in any try, P(A). P(A) does not depend on whether or not event Bย has happened. B does not influence A.

For independent events A and B, 
P(A|B) = P(A)

From lesson 9, conditional probability rule, we know that

P(A|B) = P(A โˆฉ B)/P(B)

We can combine these two and come up with a property for independent events.

P(A โˆฉ B) = P(A).P(B)

For independent events, the probability of both happening (A and B) is the product of the individual probabilities.

Let us apply this property to our example. What is the probability of not seeing a double in three consecutive rolls (with prayer ๐Ÿ™‚ or without prayer)? In other words, what are the odds of missing three rounds of the game and paying $50 to get my freedom finally?

The probability of not seeing a double in any try is 30/36. 30 non-double outcomes in 36 possibilities. Since the events are independent, the likelihood of seeing three non-doubles is (30/36)(30/36)(30/36) โ‰… 58%.

I should have known that before praying.

If the events are independent, they do not influence each other. A coin toss cannot affect a dice. Torrential rain in London may have nothing to do with the severe drought in California. Your actions may not influence my actions because we are independent.

We all like being independent … or the illusion of independence!

 

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