Question

1. Suppose a fair coin is flipped 64 times. Find a bound on the probability that the number of times the coin lands on heads is at least 24 or at most 40.

Answer 1

P( Head ) = 0.5 = p

The coin is flipped 64 times, n= 64

Let X be the number of times the coin lands on heads

X~ Binomial ( 64, 0.5)

Since , n is large enough we will use normal approximation

X~ Normal ( np, np(1-p))

X~ Normal ( 32, 16)

P( 24 <= X <= 40)= P ( 23.5 < X < 40.5) ; continuity correction

= P( << )

= P( -2.12 < z < 2.12)

= P( z < 2.12) - P( z < -2.12)

= P( z < 2.12) - 1 + P(z < 2.12)

= 2 * P( z <2.12) - 1

= 2 *0.98300 - 1

= 0.966

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