Question

Q.1 (25') Pony is playing coin tossing game with Yanny. They found the coin have 4 heads and 6 tails in 10 flips. Let p be the probability for obtaining a head, based on the first 10 flips a) Can we conclude it is a biased or fair coin base on the result above? b) Plot the Bernoulli's PMF What is the probability for obtaining 6 heads in 10 flips using the same coin? d) What is the probability for obtaining the first head in the 5th flip? eYanny further observe the coins for 10 more flips, with a total of 11 heads and 9 tails in 20 flips. Suppose all flips are random and independent events, what is the probability for 14 heads in 25 flips by the latest statistics? Hint: use the Poisson random variable, and you need to compute λ first)

Answer 1

Given that a coin is tossed n=10 times and observed 4 heads.

Thus Probability of Head = P[Head] = 0.4

And P[Tail] = 0.6

1. Let us set up the null hypothesis as

H0: Coin is Unbiased or P[H]=P[T] = 0.5 Vs HA: P[H]P[T]

To test this hypothesis we have test statistic given by;

Z = n * (Sample Proportion – P)/P*(1-P) ~ N(0,1)

= 10 * (0.4 – 0.5)/.5*0.5)

=

|Z| = 5.6921 > Z table= 1.96 or 2.58 (At both 5% and 1% LOS)

Hence H0 is rejected indicating that coin may be treated as biased coin.

b) c)

Pmf of Bernouli 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.6 0.4 Head Tail

1. Define random variable X: Number of heads in n=10 tosses.

X: 0,1,2,3,4,5,6,7,8,9,10.

Here p=P[H]=0.4

Therefore X ~ B(n=10, p=0.4)

Probability of obtaining 6 heads in 10 flips = P[X = 6] =

d) Probability of obtaining 1^st head in the 5^th flip = P[T]⁴*P[H] = 0.6⁴*0.4 = 0.05184

e)

If 10 more flips gives 11 heads in all, then Probability of Head = P[H] = 9/20 = 0.45

Our interest is to find the probability of getting 14 heads in 25 flips.

Assume that n = 25 is large and use Poisson distribution

Here X ~ P(λ = np) i.e X ~ P(λ = 25 * 0.45) i.e. X ~ P(λ = 11.25)

Required probability is P[X = 14] = 0.077609