Respuesta :
Answer:
(a) The value of P (X = 4) is 0.1465.
(b) The probability that all 5 of them land heads up is 0.0313.
(c) The probability that the student answers correctly on less than 4 of these questions is 0.6980.
Step-by-step explanation:
A Binomial distribution is the probability distribution of the number of successes, X in n independent trials with each trial having an equal probability of success, p.
The probability mass function of a Binomial distribution is:
[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0,1,2,3...,\ 0<p<1[/tex]
(1)
The information provided is:
X = 4
n = 8
p = 0.31
Compute the value of P (X = 4) as follows:
[tex]P(X=4)={8\choose 4}0.31^{4}(1-0.31)^{8-4}\\=70\times 0.00923521\times 0.22667121\\=0.146535\\\approx 0.1465[/tex]
Thus, the value of P (X = 4) is 0.1465.
(2)
The probability of heads, on tossing a single fair coin is, p = 0.50.
It is provided that n = 5 fair coins are tossed together.
Compute the probability that all 5 of them land heads up as follows:
[tex]P(X=5)={5\choose 5}0.50^{5}(1-0.50)^{5-5}\\=1\times 0.03125\times 1\\=0.03125\\\approx 0.0313[/tex]
Thus, the probability that all 5 of them land heads up is 0.0313.
(3)
There are 5 possible answers for every multiple choice question. Only one of the five options is correct.
The probability of selecting the correct answer is, p = 0.20.
Number of multiple choice questions, n = 14.
Compute the probability that the student answers correctly on less than 4 of these questions as follows:
P (X < 4) = P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)
[tex]=\sum\limits^{3}_{x=0}{{14\choose x}0.20^{x}(1-0.20)^{14-x}}\\=0.0439+0.1539+0.2501+0.2501\\=0.6980[/tex]
Thus, the probability that the student answers correctly on less than 4 of these questions is 0.6980.
