Five fair coins are tossed simultaneously Find the probabili

Five fair coins are tossed simultaneously. Find the probability function of the random variable X equals number of heads and compute the following. (You may use binomial distribution). The probability of obtaining no heads. The probability of obtaining precisely 1 head. The probability of obtaining at least 1 head . The probability of obtaining no more than 3 heads.

Solution

4.

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 =    5      
p = the probability of a success =    0.5      

Thus,

P(5, x) = 5Cx * (0.5^x) * (0.5^(n-x))

P(5, x) = 5Cx * (0.5^n) [probability distirbution]

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a)

If

x = the number of successes =    0      

          
Thus, the probability is          
          
P (    0   ) =    0.03125 [ANSWER]

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b)

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 =    5      
p = the probability of a success =    0.5      
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.15625 [ANSWER]

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c)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    5      
p = the probability of a success =    0.5      
x = our critical value of successes =    1      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   0   ) =    0.03125
          
Thus, the probability of at least   1   successes is  
          
P(at least   1   ) =    0.96875 [ANSWER]

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d)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    5      
p = the probability of a success =    0.5      
x = the maximum number of successes =    3      
          
Then the cumulative probability is          
          
P(at most   3   ) =    0.8125 [ANSWER]