Question 1: A user has beaten 4192 out of 6378 levels attempted on a game. If th

Question

Question 1: A user has beaten 4192 out of 6378 levels attempted on a game. If the user wants to beat 6 levels on a given day, what is the probability they will finally do so while playing their 13th level?

Question 2: A user has a 55% chance of beating a level first. Another user has a 45% chance of beating a level first. If they race to be the first person to beat 5 levels, what is the probability it will take exactly 8 levels before someone gets 5 wins? note: Results of each level are independent.

Explanation / Answer

1. Probability of winning levels = 4192/6378 = 0.6572

We need to know the probability of winning the 6th level in the 13th level. This is the case of negative binomial experiment which results in 5 successes after level 12 and 6 successes after level 13.

b*(x; r, P) = x-1Cr-1 * Pr * (1 - P)x - r

b*(13; 6, 0.6572) = C(12,5) * 0.6572^6 * (1-0.6572)^7

= 792 * 0.6572^6 * 0.3428^7

= 0.0355

Probability of winning 6 levels in the 13th level = 0.0355

2. Probability of winning 5 levels in the 8th level for 1st user =

b*(8; 5, 0.55) = C(7,4) * 0.55^5 * (1-0.55)^3

= 0.1605

Probability of winning 5 levels in the 8th level for 2nd user =

b*(8; 5, 0.45) = C(7,4) * 0.45^5 * (1-0.45)^3

= 0.1074

As the probability of the 1st user is greater than the 2nd user, the probability it will take exactly 8 levels before someone gets 5 wins = 0.1605

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