Suppose you flip eight fair coins. What is the probability of flipping exactly eight heads?What is the probability of flipping at least four heads?What is the probability of flipping at most seven heads?Please do not solve using Microsoft Excel.  My professor is not teaching through excel and I would like to see the steps and formula for solving the problem.

Question
Asked Oct 26, 2019
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  1. Suppose you flip eight fair coins.

 

  1. What is the probability of flipping exactly eight heads?
  2. What is the probability of flipping at least four heads?
  3. What is the probability of flipping at most seven heads?

Please do not solve using Microsoft Excel.  My professor is not teaching through excel and I would like to see the steps and formula for solving the problem.

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Expert Answer

Step 1

Binomial distribution:

The events that has exactly two possible outcomes (success or failure) follows binomial distribution with parameter n and p. The probability mass function of binomial distribution is as follows:

Р(х - х) - "С, р" (1-р)"*
where, p is the probability of success
n is the number of trials
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Р(х - х) - "С, р" (1-р)"* where, p is the probability of success n is the number of trials

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Step 2
  1. Computation of probability of flipping exactly eight heads:

Here, n=8

The probability of getting heads in a fair coin is 0.5 (=1/2).

The probability of flipping exactly eight heads is obtained as 0.0039, from the calculation given below:

P(x=8)= C,(0.5) (1-0.5)6-9)
8!
x(0.5)'x (0.5)
8!(8-8)
=0.0039
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Image Transcriptionclose

P(x=8)= C,(0.5) (1-0.5)6-9) 8! x(0.5)'x (0.5) 8!(8-8) =0.0039

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Step 3
  1. Computation of probability of flipping at least four heads:

The probability of flipping at least four heads is ...

P(x24)-1-[P(x-0) + P(x=1)+P(x=2)+ P(x=3)]
+
+ °C(0.5)(1-0.5)e-)
C(0.5) (1-0.5)e-)
C(0.5)'(1-0.5)6-0)
=1-
C,(0.5) (1-0.5)-)
;x (0.5) x (0.5)10
8!
8!
; x (0.5) x(0.5)
1!(8-1)!
0!(8-0)!
=1-
(0.5) x(0.5)|318.
8!
8!
x (0.5 x(0.5)
3!(8-3)
2!(8-2)
1-10.0039 0.0313 0.1094 +0.2188]
=1-0.3633
=0.6367
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P(x24)-1-[P(x-0) + P(x=1)+P(x=2)+ P(x=3)] + + °C(0.5)(1-0.5)e-) C(0.5) (1-0.5)e-) C(0.5)'(1-0.5)6-0) =1- C,(0.5) (1-0.5)-) ;x (0.5) x (0.5)10 8! 8! ; x (0.5) x(0.5) 1!(8-1)! 0!(8-0)! =1- (0.5) x(0.5)|318. 8! 8! x (0.5 x(0.5) 3!(8-3) 2!(8-2) 1-10.0039 0.0313 0.1094 +0.2188] =1-0.3633 =0.6367

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