The trick I am using to solve this question was taught to me by a channel on YouTube named Salman Gaffar. It’s Probability 07

So first let’s list down the possible outcomes

1) H H H H H H
2) H H H H H T
3) H H H H T T
4) H H H T T T
5) H H T T T T
6) H T T T T T
7) T T T T T T

We want to find out the probability of Atleast 2 tails and atmost 5 tails so option 1,2,7 are unfavourable and 3,4,5,6 are favourable

We can find probability of either one of them. Remember if we find probability of unfavourable outcomes we need to subtract it by 1 to get the probability of favourable outcomes. Let me show how

Let’s find the probability of option 1

1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/64

For option 2

1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/64 x 6!/5! = 6/64

For option 7

1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/64


So it’s either option 1 OR option 2 OR option 7 as unfavourable outcome

Since it’s OR we add, if its AND we multiply

1/64 + 1/64 + 6/64 = 8/64 or 1/8

1 - 1/8 = 7/8

Option C is the right answer

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We need to determine the following probabilities:

TTHHHH, TTTHHH, TTTTHH, and TTTTTH

P(TTHHHH) = (1/2)^6 = 1/64

TTHHHH can be arranged in 6C2 = 6!(2! x 4!) = (6x5)/2 = 15 ways, so the probability is 15/64.

P(TTTHHH) = (1/2)^6 = 1/64

TTTHHH can be arranged in 6C3 = 6!(3! x 3!) = (6x5x4)/3! = 20 ways, so the probability is 20/64.

P(TTTTHH) = (1/2)^6 = 1/64

TTTTHH can be arranged in 6C4 = 6!(2! x 4!) = (6x5)/2 = 15 ways, so the probability is 15/64.

P(TTTTTH) = (1/2)^6 = 1/64

TTTTTH can be arranged in 6C5 = 6!/5! = 6 ways, so the probability is 6/64.

Thus, the total probability is (15 + 20 + 15 + 6)/64 = 56/64 = 7/8.

Answer: C
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The opposite even would be 0, 1, or 6 tails.

\(P(t = 0) = P(h = 6) = (\frac{1}{2})^6\)

\(P(t = 1) = \frac{6!}{5!}*(\frac{1}{2})*(\frac{1}{2})^5\). We are multiplying by 6!/5! because thhhhh can occur in 6!/5! = 6 different ways: thhhhh, hthhhh, hhthhh, hhhthh, hhhhth, hhhhht.

\(P(t = 6) = (\frac{1}{2})^6\)


\(P (t = 2, 3, 4, \ or \ 5) = 1- P(t = 0, 1, \ or \ 6) = 1 - ((\frac{1}{2})^6 + \frac{6!}{5!}*(\frac{1}{2})*(\frac{1}{2})^5 + (\frac{1}{2})^6) = \frac{7}{8}\)


Answer: C.
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