I am trying to improve my stat skills from a book that gives an exercise, I cannot get my head around.&lt;br.It goes like this:&lt;br.There are 5 green balls and 1 red ball in a box. We draw four times randomly with replacement. What is the probability that we draw at least two red balls?&lt;br.My guess would be&lt;br.(1/6)^2 + (1/6)^3 + (1/6)^4 because the probability of at least 2 red balls out of 4 draws must be equal to drawing two or three or four times a red ball. Hence I would use the addition rule, which gives 0.03 = 3 percent&lt;br.However, the book says it is 13 percent, and gives the following explanation (150+20+1)/1296.&lt;br.Why is this so?&lt;br.My attempt to trace back the terms:&lt;br.I can possibly see how it got the first and third term in the fraction: The first could be (5/6)^2 = 25/36, which is the probability of drawing two green balls; and the third could be (1/6)^4 = 1/1296, which is the probability of drawing four red balls. I have no idea though where they could possibly get the 20 from.

Question

I am trying to improve my stat skills from a book that gives an exercise, I cannot get my head around.&amp;lt;br.It goes like this:&amp;lt;br.There are 5 green balls and 1 red ball in a box. We draw four times randomly with replacement. What is the probability that we draw at least two red balls?&amp;lt;br.My guess would be&amp;lt;br.(1/6)^2 + (1/6)^3 + (1/6)^4 because the probability of at least 2 red balls out of 4 draws must be equal to drawing two or three or four times a red ball. Hence I would use the addition rule, which gives 0.03 = 3 percent&amp;lt;br.However, the book says it is 13 percent, and gives the following explanation (150+20+1)/1296.&amp;lt;br.Why is this so?&amp;lt;br.My attempt to trace back the terms:&amp;lt;br.I can possibly see how it got the first and third term in the fraction: The first could be (5/6)^2 = 25/36, which is the probability of drawing two green balls; and the third could be (1/6)^4 = 1/1296, which is the probability of drawing four red balls. I have no idea though where they could possibly get the 20 from.

wasipewelr · Accepted Answer

The probability of getting exactly 2 red balls is            (              1        6            )        2              (              5        6            )        2                      (                    4      2                      )              =      25    1296    ⋅  6where the binomial coefficient is just the number of ways you can select which two picks, out of the four, got red balls.The probability of getting exactly 3 red balls is            (              1        6            )        3              (              5        6            )        1                      (                    4      1                      )              =      5    1296    ⋅  4where the binomial coefficient is just the number of ways you can select which one pick, out of the four, got a red ball. That is where the 20 comes from.Can you take it from here?

I am trying to improve my stat skills from a book that gives an exercise, I cannot get my head aroun

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