A fair coin tossed three times and the events A B and C are

A fair coin tossed three times and the events A, B, and C are defined as follows:

A: {Atleast one head is observed}

B: {Atleast two heads are observed}

C: {The number of heads observed are odd}

Find the following probabilities by summing the probabilities of the appropriate sample points

(a) P(B):_________

Solution

For event A:

P(at least one head) = 1 - P(no heads)

= 1 - (1/2)(1/2)(1/2)

P(at least one head) = 7/8 = 0.875

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For event B:

P(at least 2 heads) = P(2 heads) + P(3 heads)

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 heads   ) =    0.375

Similarly,

P (    3 heads) =    0.125

Thus,

P(at least 2 heads) = 0.375 + 0.125 = 0.5

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P(odd) = P(1) + P(3)

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 =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.375

As we saw, P(3) = 0.125.

Thus,

P(odd) = P(C) = 0.375 + 0.125 = 0.50

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Summary:

P(A) = 0.875
P(B) = 0.5
P(C) = 0.5

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PART A)

P(B) = 0.5

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PART B)

P(A U Bc) = P(at least 1 head or \"not at least 2 heads\")

= P(1 head)

As we saw earlier, this is

= 0.375 [answer]

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PART C:

P(at least one head AND at least 2 heads AND odd heads)

= P(3 heads)

As we saw earlier, this is

= 0.125 [answer]