Question

A radio tube inserted into a system has probability 0.2 of lasting 500 hours. 20 tubes are tested.

1. Find probability that exactly that 'k' of these tubes will last more than 500 hours.

2. Find probability that number of tubes that last more than 500 hours will fall between 12 and 17.

3. Sketch the CDF of the random variable that describes the random phenomenon

Also: a die is rolled 120 times. Find probability that 35 or more sixes will be rolled.

Show all steps, thank you.

Answer 1

1)

X = number tubes which last more than 500 hours

X follow binomial distribution with n = 20 , p = 0.2

P(X= k) = nCk p^k (1-p)^(n-k)

= 20Ck 0.2^k * 0.8^(20-k)

2)

P(12 <= X <= 17)   {i am taking both inclusive}

= P(X<= 17) - P(X<= 11)

=

0.000102

3)

x	p	cdf
0	0.011529	0.011529
1	0.057646	0.069175
2	0.136909	0.206085
3	0.205364	0.411449
4	0.218199	0.629648
5	0.17456	0.804208
6	0.1091	0.913307
7	0.05455	0.967857
8	0.022161	0.990018
9	0.007387	0.997405
10	0.002031	0.999437
11	0.000462	0.999898
12	8.66E-05	0.999985
13	1.33E-05	0.999998
14	1.66E-06	1
15	1.66E-07	1
16	1.3E-08	1
17	7.65E-10	1
18	3.19E-11	1
19	8.39E-13	1

Excel formula

x	p	cdf
0	=BINOMDIST(A2,20,0.2,0)	=B2
=1+A2	=BINOMDIST(A3,20,0.2,0)	=C2+B3
=1+A3	=BINOMDIST(A4,20,0.2,0)	=C3+B4
=1+A4	=BINOMDIST(A5,20,0.2,0)	=C4+B5
=1+A5	=BINOMDIST(A6,20,0.2,0)	=C5+B6
=1+A6	=BINOMDIST(A7,20,0.2,0)	=C6+B7
=1+A7	=BINOMDIST(A8,20,0.2,0)	=C7+B8
=1+A8	=BINOMDIST(A9,20,0.2,0)	=C8+B9
=1+A9	=BINOMDIST(A10,20,0.2,0)	=C9+B10
=1+A10	=BINOMDIST(A11,20,0.2,0)	=C10+B11
=1+A11	=BINOMDIST(A12,20,0.2,0)	=C11+B12
=1+A12	=BINOMDIST(A13,20,0.2,0)	=C12+B13
=1+A13	=BINOMDIST(A14,20,0.2,0)	=C13+B14
=1+A14	=BINOMDIST(A15,20,0.2,0)	=C14+B15
=1+A15	=BINOMDIST(A16,20,0.2,0)	=C15+B16
=1+A16	=BINOMDIST(A17,20,0.2,0)	=C16+B17
=1+A17	=BINOMDIST(A18,20,0.2,0)	=C17+B18
=1+A18	=BINOMDIST(A19,20,0.2,0)	=C18+B19
=1+A19	=BINOMDIST(A20,20,0.2,0)	=C19+B20
=1+A20	=BINOMDIST(A21,20,0.2,0)	=C20+B21