A gambler decides there are enough people who would be tempted to play toss-a-co

Question

A gambler decides there are enough people who would be tempted to play toss-a-coin and decides to con them by having two coins: a fair coin which lands heads and tails with equal probability and a biased coin which lands heads twice as often on average as it lands tails. He draws people in by asking them to bet on the number of heads in many tosses. What is the probability of getting a head if he tosses the fair coin once? What is the probability of getting a head if he tosses the biased coin once? What is the probability of getting 2 heads and a tail if he tosses the fair coin thrice? Find the same for the biased coin. Suppose he tosses one of the two coins (fair or unbiased) thrice and you are given that he gets 2 heads and a tail. What is the probability that he used the fair coin for the three tosses? What is the probability that he used the biased coin?

Explanation / Answer

Let

2m = probability of head for a biased coin
m = probability of tails for a biased coin

hence,

2m + m = 1

3m = 1

m = 1/3

Hence,

P(head) = 2/3.

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For the fair coin, the probability of a head is

p = 0.5

while for the biased coin, the probability of a head is

p = 2/3 or 0.6666666667

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a)

fair:
P = 0.5 [ANSWER]

biased:
P = 2/3 or 0.6666666667 [ANSWER]

**********************************************************************************

b)

fair coin:

P(3 heads) = 0.5^3 = 0.125 [ANSWER]

biased:

P(3 heads) = (2/3)^3 = 8/27 or 0.296296296 [ANSWER]

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c)

For the fair coin:

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    3      
p = the probability of a success =    0.5      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) = P(2heads|fair) =   0.375

****
For the biased coin:

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    3      
p = the probability of a success =    0.666666667      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) = P(2 heads|biased) =   0.444444444
******

Hence,

P(2 heads) = P(fair) P(2 heads|fair) + P(biased)P(2 heads|biased)

P(2 heads) = (1/2)*(0.375) + (1/2)*0.444444444 = 0.409722222

Hence,

P(fair|2 heads) = P(fair)P(2 heads|fair)/P(2 heads) = (1/2)*0.375/0.409722222 = 0.457627119 [ANSWER]

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Meanwhile, P(biased|2 heads) = 1 - P(fair|2 heads) = 0.542372881 [ANSWER]

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