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Let X represents the difference between the number of heads and the number of tails
when a coin is tossed 33 times. Then P(X=3)=
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In this problem, the sum Heads+Tails is 33, while the difference Heads-Tails = 3.
It immediately follows that Heads = 18, Tails = 15.
Therefore, the event X= 3 is EQUIVALENT to event Tails = 15.
Now, P(Tails=15) is (use the formula of probability for the Binomial distribution)
P(Tails=15) = = BINOM.DIST(15, 33, 0.5, FALSE) = 0.120741. ANSWER
I used the standard EXCEL function to calculate the binomial distribution probability.
Its second parameter is the "number of trials n" (n= 33 in this case);
the first parameter is the "number of successful trials" (k= 15 in this case);
third parameter is "the probability of success in each one single trial" (p= 0.5 in this case); and
the fourth parameter says if you want calculate a single addend or the sum of addends (the single addend in this case).
Input "=BINOM.DIST(15, 33, 0.5, FALSE)" into any MS Excel cell in your computer and press enter to get the answer P = 0.120741 immediately.
On Excel function BINOM.DIST, see its description everywhere, for example
https://support.office.com/en-us/article/binom-dist-function-c5ae37b6-f39c-4be2-94c2-509a1480770c
Solved.