If an unfair coin is flipped 16 times, the probability of getting at

Question

If an unfair coin is flipped 16 times, the probability of getting at least one head is 15/16. What is the probability of getting at least one head if the coin is flipped 8 times?

A. 1/16
B. 1/8
C. 1/4
D. 1/2
E. 3/4

Bunuel · Accepted Answer

Official Solution:

If an unfair coin is flipped 16 times, the probability of getting at least one head is  \frac{15}{16} . What is the probability of getting at least one head if the coin is flipped 8 times?

A.  \frac{1}{16} 
B.  \frac{1}{8} 
C.  \frac{1}{4} 
D.  \frac{1}{2} 
E.  \frac{3}{4}

A fair coin has the probability of heads equal to the probability of tails, so  P(tail) = P(head) = \frac{1}{2} . We are told that the coin is unfair, so  P(tail) ≠ P(head) .

Let's denote the probability of getting tails as  x  and the probability of heads as  1−x . In this case, the probability of getting at least one head in 16 flips would be  P(heads \geq 1) = 1 - P(tails = 16) = 1 - x^{16} . Given that this equals  \frac{15}{16} , we have:

1 - x^{16} = \frac{15}{16}

x^{16} = \frac{1}{16}

The question asks to find the probability of getting at least one head if the coin is flipped 8 times, which would equal  P(heads \geq 1) = 1 - P(tails = 8) = 1 - x^{8} . Since  x^{16} = \frac{1}{16} , taking the square root would give  x^{8} = \frac{1}{4} . Therefore,  1 - x^{8} = 1 - \frac{1}{4} = \frac{3}{4}

Answer: E

gmatophobia · Answer

Assume in one flip

Probability(Tail) = x

Probability of getting atleast one head when the coin is flipped 16 times = 1 - Probability of getting all tails when the coin is flipped 16 times

Probability of getting all tails when the coin is flipped 16 times =  (	ext{Probability(Tail)})^{16}

\frac{15}{16} = 1- (x)^{16}

\frac{1}{16} = (x)^{16}  ➔ Equation 1

Probability of getting atleast one head when the coin is flipped 8 times = 1 - Probability of getting all tails when the coin is flipped 8 times

Probability of getting all tails when the coin is flipped 8 times =  (	ext{Probability(Tail)})^{8}

(	ext{Probability(Tail)})^{8} = ((	ext{Probability(Tail)})^{16})^{\frac{1}{2}}

(	ext{Probability(Tail)})^{8} = (x^{16})^{\frac{1}{2}}

Taking square root of equation (1) we get

x^{8} = \frac{1}{4}

Probability of getting all tails when the coin is flipped 8 times =  (x)^{8} = \frac{1}{4}

Probability of getting atleast one head when the coin is flipped 8 times =  1 - \frac{1}{4}

=  \frac{3}{4}

Option E

sayan640 · Answer

Probability of getting All tails = (Prob. of tails) ^ (no of tails) * Prob. of head ^(no of heads ) * total no of toss c no of tails
= x^16 * anything^0 *16 c 16
= x^16
Hence , x^16 = 1/16

hotshot68 · Answer

When a coin is tossed for 16 times , total no of outcome =2^16
P(At least 1 head ) = 1 -P(no heads)
15/16 = 1-P(no heads)
P(no heads)= \frac{1}{16} 
Also P(no heads ) =  \frac{No. of outcomes .as. no. head }{ Total . no. of outcomes} 
 \frac{1}{16}  =  \frac{x}{2^16} 
x= 2^12

When coin is tossed 8 times , total no of outcome =  2^8 
P(no head) = \frac{y}{2^8} 
Now  2^8  = (2^16)^\frac{1}{2} 
so y =  (2^12)^\frac{1}{2}  = 2^6 
P(no head) = \frac{ 2^6}{2^8}  = \frac{1}{4} 
P(atleast one head ) =1- \frac{1}{4} = \frac{3}{4}

If an unfair coin is flipped 16 times, the probability of getting at

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