![]() Today my distraction came in the form of a Tweet by David Robinson demonstrating how flipping a coin and getting a heads and then another heads takes 6 flips on average while a heads then a tails only takes 4.Ī #tidyverse simulation to demonstrate that if you wait for two heads in a row, it takes 6 flips on average, while you wait for a heads then a tails, it takes 4 flips on average Plus, it’s a dissertation distractions are welcome in any flavor. Don’t get me wrong, I find studying hate speech very fascinating, but in all honesty, it gets to be a bit much sometimes. I’m writing up my dissertation…but occasionally I need a distraction. Thus the probability of a run of exactly 6 heads is. “Figure 2 shows that the probability of a run of at least 7 heads is. I disagree with the final value in this statement I liked the Excel approach with “states” being the number of Heads (success) remaining instead of one where the states are the # of Heads in a row (2007) Understanding probability: chance rules in everyday life (2 nd ed.). The probability of getting a run of at least 6 heads or tails in 20 tosses of a fair coin is 23.88% (cell U23). We solve Example 2 by building the Excel worksheet shown in Figure 3.įigure 3 – Run of at least 6 in 20 tosses Thus the probability that there will be a run of at least r in n tosses of a fair coin is given by g( n – 1, 1). Here the leftmost term on the right side of the equation corresponds to getting the same outcome as on the previous toss when i tosses remain and the rightmost term corresponds to getting a different outcome from the previous toss when i tosses remain. 5.įor all 0 < i < n and j < r, we have the following recursive formula: This time to keep things simple, we will assume that we have a fair coin and so p =. G( i, j) = the probability of getting a run of least r during n tosses assuming there are i tosses remaining and the last j tosses have all been the same and so far there has not been a run of r heads or tails. ![]() Once again we solve this problem by recursion. Here the run can be of either heads or tails. Run of heads or tailsĮxample 2: What is the probability that there will be a run of at least 6 in 20 tosses of a fair coin? ![]() 064133 since we can have situations where there are runs of 6 heads as well as 7 or more heads (e.g. We can’t say that the probability of a run of exactly 6 heads is. Thus the probability that the longest run of heads is exactly 6 heads is. If we want to know the probability that the longest run of heads in 20 tosses is 6 heads, then we need to first calculate the probability of a run of at least 7 heads in 20 tosses, as shown in Figure 2.įigure 2 – Run of at least 7 heads in 20 tossesĬell K24 shows that the probability of a run of at least 7 heads is 5.82%. Therefore, the probability of getting a run of at least 6 heads in 20 tosses of a fair coin is 12.23%. 122315, which is the result we are looking for. We see (cell B24) that the value of f(20, 0) =. We then copy this formula into the rest of the table by highlighting the range B5:G24 and pressing Ctrl-D and Ctrl-R. Here range B3:G3 consists of all 0’s, range H4:H24 consists of all 1’s. We solve Example 1 by building an Excel worksheet as shown in Figure 1.įigure 1 – Run of at least 6 heads in 20 tosses The probability that there will be a run of at least r heads in n tosses of a coin with probability p that heads will occur on any toss is given by f( n, 0). Here the leftmost term on the right side of the equation corresponds to getting heads when i tosses remain and the rightmost term corresponds to getting tails when i tosses remain. Now for i = 0, 1, …, n, and j = 0, 1, …, r defineį( i, j) = the probability of getting a run of at least r heads in n tosses assuming there are i tosses remaining and the last j tosses have all been heads and so far there has not been a run of r heads.Ĭlearly for all j 0 and j < r, we have the following recursive formula:į( i, j) = p ∙ f( i – 1, j + 1) + (1 – p ) ∙ f( i – 1, 0) Let p = the probability that heads will occur on any toss, r = the size of the run we are looking for and n = the total number of tosses. Example 1: What is the probability that there will be a run of at least 6 heads in 20 tosses of a fair coin?
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