Question

Calculate the following binomial probability by either using one of the binomial probability tables, software, or a calculator using the formula below. Round your answer to 3 decimal places.

P(x | n, p) =

n!

(n − x)! x!

· p^x · q^{n − x}    where    q = 1 − p

P(x < 19, n = 20, p = 0.9)

=

Answer 1

P(x < 19, n = 20, p = 0.9) = 0.608

The complete solution of above answer are as below

The values provided in the above question are

n = 20,

p = 0.9

The formula of binomial probability is

  

We have to find the probability of  P(x < 19, n = 20, p = 0.9)

We have to find the probability of ​​​​​​ that is

We find the above probability using following Excel function

=BINOMDIST(number_s,trials,probability_s,cumulative)

We choose following value for above Excel function

number_s = 18,  trials = 20,   probability_s = 0.9, cumulative = TRUE-cumulative distribution function

Using the above values Excel function becomes

=BINOMDIST(18,20,0.9,TRUE-cumulative distribution function) and then press Enter

we get the above P(x < 19, n = 20, p = 0.9)  are as below

=0.608253 0.608 (Round your answer to 3 decimal places)

P(x < 19, n = 20, p = 0.9) = 0.608

Summary :-

P(x < 19, n = 20, p = 0.9) = 0.608

The following table is only for your understanding

We find following table value using Excel function are as below

1) For, P(x) we use =BINOMDIST(number_s,trials,probability_s,cumulative)

The value of number_s = 0,1,2,............,20 We choose one value at a time from 0,1,2,............,20

trials = 20 ,  probability_s = 0.9, cumulative = FALSE-probability distribution function

We get the values of P(x) from 0,1,2,.............,20 in second column in following table

2) For, P(x) we use =BINOMDIST(number_s,trials,probability_s,cumulative)

The value of number_s = 0,1,2,............,20 We choose one value at a time from 0,1,2,............,20

trials = 20 ,  probability_s = 0.9, cumulative = TRUE-cumulative distribution function

We get the values of F(x) from 0,1,2,.............,20 in third column in following table

The table of P(x) and F(x) is as below

When We have to find P(x < 19, n = 20, p = 0.9) then in second column we add P(x) from x = 0,1,2,.........,18

then we get P(x < 19, n = 20, p = 0.9) =0.608253 0.608 (we need our answer up to 3 decimal places)

Or

We find P(x < 19, n = 20, p = 0.9) using F(x) then use only following command to get our answer

=BINOMDIST(18,20,0.9,TRUE-cumulative distribution function) and then press Enter

P(x < 19, n = 20, p = 0.9) =0.608253 0.608 (we need our answer up to 3 decimal places)

x	P(x)	F(x)
0	1E-20	1E-20
1	1.8E-18	1.81E-18
2	1.54E-16	1.56E-16
3	8.31E-15	8.47E-15
4	3.18E-13	3.26E-13
5	9.15E-12	9.48E-12
6	2.06E-10	2.15E-10
7	3.71E-09	3.92E-09
8	5.42E-08	5.81E-08
9	6.51E-07	7.09E-07
10	6.44E-06	7.15E-06
11	5.27E-05	5.99E-05
12	0.000356	0.000416
13	0.00197	0.002386
14	0.008867	0.011253
15	0.031921	0.043174
16	0.089779	0.132953
17	0.19012	0.323073
18	0.28518	0.608253
19	0.27017	0.878423
20	0.121577	1