Question

Suppose that a classroom has 8 light bulbs. The probability that each individual light bulb works is 0.8. Suppose that each light bulb works independently of the other light bulbs. What is the probability that all eight of the light bulbs work?

a.0.17
b.0.13
c.0.00001024
d.0.8

Answer 1

Answer:

a. 0.17

Step-by-step explanation:

Total number of light bulbs = 8

The probability that each individual light bulb works = 0.8

The working of light bulbs is independent of each other, this means one light bulb does not influence the other light bulbs.

We need to calculate the probability that all eight of the light bulbs work. Since the light bulbs work independently, the overall probability of independent events occurring together is the product of their individual probabilities. Therefore,

Probability that all eight of the light bulbs work = 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8

= $(0.8)^{8}$

= 0.16777216

≈ 0.17

Thus, option a gives the correct probability that all eight of the light bulbs work

Answer 2

You can use binomial distribution, and thus, its probability function to find the needed probability.

The probability that all eight of the light bulbs work is 0.167

How to find that a given condition can be modeled by binomial distribution?

Binomial distributions consists of n independent Bernoulli trials.

Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))

Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as

$X \sim B(n,p)$

The probability that out of n trials, there'd be x successes is given by

$P(X =x) = \: ^nC_xp^x(1-p)^{n-x}$

Using the above method to find the needed probability

Since all the light bulbs' working is independent, and each bulb's chance of working is 0.8 and there are 8 bulbs, thus,

n = 8

p = 0.8

and Let X be a random variable tracking how many out of 8 bulbs are working, then we have:

$X \sim B(8, 0.8)$

Then, the needed probability is P(X = 8) (since we need to know probability that all 8 bulbs will work)

By using the probability mass function of binomial distribution, we get:

$P(X =x) = \: ^nC_xp^x(1-p)^{n-x}\\P(X = 8) = \:^8C_8(0.8)^8(1-0.8)^{8-8} = 1 \times (0.8)^8 \times 1 \approx 0.167$

Thus,

The probability that all eight of the light bulbs work is 0.167

Learn more about binomial distribution here:

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