a club with 29 members A club with 29 members is voting on a proposal. A group o

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a club with 29 members A club with 29 members is voting on a proposal. A group of 4 has decided to vote in favor of the proposal while each of the rest of the members vote independently with a chance of 49% in favor of the proposal, what are the chances o the proposal will get the majority vote 0 Check You toss a coin 208 times. Find the probability that exactly 104 of the 208 tosses will be heads. Check .00 About 4% of the scientists in the National Academy of Sciences believe in PSI (parapsychology such as telepathy, and astral projection). At a convention 191 members show up. Let X be the number of scientists at the convention who believe parameters n = 191 and p = 0.04. in PSI. You may assume for the following questions that X is binomially distributed with f 4.00 (a) How many scientists at the convention do you expect do believe in PS1i? (b) What are the chances that none of the scientists at the convention believes in PSI? (c) What are the chances that at least 4 scientists at the convention believe in PS1? (d) What is the standard deviation of X? tion

Explanation / Answer

1) 4 members out of 29 members have decided to vote in favour. To achieve the majority, at least 15 votes are needed out of 29 votes. Since 4 votes are in favour, we have to find the probability that at lest 11 votes are in favour out of remaining 25 votes. The probability of the vote in favour p=0.49 for each voters who are independent. So, the number of votes (X) in favour out of 25 voters is a random variable which follows a Binomial distribution with parameters n=25 and p=0.49. Then the required probability is,

P(X>=11) = 0.7574456 (Using R-code: 1-pbinom(10,25,0.49) )

The probability can also be approximated by a normal probability as follows:

P(X>=11) = P( (X-E(X))/s(X) >= ((10.5 - E(X))/s(X)) ) = P(z >=  -0.70014 ) = 1-Phi(-0.70014) = 0.75808.

2) If the coin is unbiased, then the probability of head, p=0.5. Then probability of occurring 104 heads out of 208 tosses = 208C104*(0.5)^208 = 0.05525689. [This can be calculated in R using R-code: dbinom(104,208,0.5) ]

3) (a) Expected number of scientists who beleive in PSI = E(X) = n*p = 191*0.04 = 7.64 = 8 (Approx.)

(b) Probability of none of them beleive in PSI = P(X=0) = (0.96)^191 = 0.000411.

(c) P(X>=4) = 1 - P(X=0) - P(X=1) - P(X=2) - P(X=3) = 1 - 0.05061129 = 0.9493887

(d) standard deviation of X = sqrt(191*0.04*0.96) = 2.70821.

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