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“There are 13 Hearts”: If there are an infinite number of draws how does the probability eventually converge to 9/16?

Question

If there are an infinite number of draws how does the probability eventually converge to 9/16? There was mention in the video that there can be an infinite number of draws greater than or equal to 3 and that it would not be possible to add up the probabilities. However, through using P = 1 - P(complement) we found that the probability is 9/16. So wouldn't that mean that it would actually be possible to add up each of the probabilities of draws greater than or equal to 3? Also in the text explanation, there is mention regarding the draws being "independent". Can you help me to better understand the concept of independent?

Answer

Well, there are an infinite number of draws, but as the draw number gets higher, the probability of getting the first heart on that draw gets smaller, so that the probabilities get infinitely close to zero. The probability of getting the first heart on a certain number draw is the probability of getting the first heart on the previous draw times 3/4. So each consecutive probability (of getting a heart on a given draw) is 3/4 of the previous draw. So even though we have an infinite number of probabilities, they mathematically converge to a finite value. In this case, the probabilities of getting a first heart on the 3rd draw and beyond converge to 9/16.

It is actually possible to calculate using a sum of an infinite geometric series. You don't need to know this for the GMAT:

Sum = (first term of series)/(1 - (common ratio)) which here would be:

((3/4)^2 * (1/4))/(1 - (3/4)) = ((9/16)(1/4))/(1/4) = 9/16

As you can see, however, it's probably faster to use P = 1 - P(complement).

Independent here means that the probability of a draw is NOT affected by the outcomes before it. They are independent. I’d recommend reading the lesson on independent events.

 

Link

https://gmat.magoosh.com/questions/4224

 

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