a machine has 7 identical components which function independently. the probability that a component will fail is 0.2. the machine will stop working if more than three components fail. find the probability that the machine will be working.
Add the probabilities that 0, 1, 2 or 3 components have failed. That sum will be the probability that the machine works.
The probability that none have failed is
P(0) = 0.8^7 = 0.210
The probability that one part has failed is
P(1) = 0.8^6*0.2*7 = 0.367
The probability that two parts have failed is P(2) (0.8)^5*(0.2)^2*[7!/(5!2!)] = 0.275
The probability that three parts have failed is P(3) = (0.8)^4*(0.2)^30*[7!/4!3!)] = 0.115
The sum of these probabilities is 0.967.
thank you so very much! you're my hero! =D
Well, let's have some fun with math, shall we? To find the probability that the machine will be working, we need to figure out the probability of less than or equal to three components failing.
Now, let's break it down. The probability of a component failing is 0.2, which means the probability of a component not failing (in other words, working) is 1 - 0.2, which is 0.8. Since all the components function independently, we can calculate the probability of any specific combination occurring.
So, if no components fail, that's 0 components failing, and the probability of that happening is (0.8)^7 (because we need all 7 components to work).
If one component fails, that's 1 component failing, and the probability of that happening is (0.2)^1 * (0.8)^6 (because we need 1 component to fail and 6 components to work).
If two components fail, that's 2 components failing, and the probability of that happening is (0.2)^2 * (0.8)^5 (because we need 2 components to fail and 5 components to work).
If three components fail, that's 3 components failing, and the probability of that happening is (0.2)^3 * (0.8)^4 (because we need 3 components to fail and 4 components to work).
Now, we can add up all the individual probabilities to find the probability that the machine will be working:
P(machine working) = (0.8)^7 + (0.2)^1 * (0.8)^6 + (0.2)^2 * (0.8)^5 + (0.2)^3 * (0.8)^4
Grab your calculator and let's get crunching!
To find the probability that the machine will be working, we need to calculate the probability of having three or fewer components failing.
Since the components function independently, we can use the binomial distribution to calculate this probability.
Let's find the probability for each case:
P(no component fails) = (0.8)^7 = 0.2097152
P(one component fails) = 7C1 * (0.2)^1 * (0.8)^6 = 7 * 0.2 * (0.8)^6 = 0.33554432
P(two components fail) = 7C2 * (0.2)^2 * (0.8)^5 = 21 * 0.04 * (0.8)^5 = 0.268435456
P(three components fail) = 7C3 * (0.2)^3 * (0.8)^4 = 35 * 0.008 * (0.8)^4 = 0.057646080
Now, we can find the probability of the machine working by summing these probabilities:
P(machine working) = P(no component fails) + P(one component fails) + P(two components fail) + P(three components fail)
= 0.2097152 + 0.33554432 + 0.268435456 + 0.057646080
= 0.871340032
Therefore, the probability that the machine will be working is approximately 0.871.
To find the probability that the machine will be working, we need to find the probability that at most three components fail.
Let's break down the problem step by step:
Step 1: Find the probability of exactly zero component failing.
The probability that a component will fail is 0.2, which means the probability that a component will not fail is 1 - 0.2 = 0.8. Since all components function independently, the probability that all seven components will not fail is (0.8)^7 = 0.2097152.
Step 2: Find the probability of exactly one component failing.
The probability of one component failing is 0.2, and since there are seven components, we can choose any one of the seven to fail. Therefore, the probability of exactly one component failing is 7 * 0.2 * (0.8)^6 = 0.33554432.
Step 3: Find the probability of exactly two components failing.
The probability of two components failing is (0.2)^2, and we can choose any two out of the seven components, so we need to multiply by the number of possible combinations. The number of combinations to choose two out of seven is given by the binomial coefficient (7 choose 2), denoted as C(7, 2), which is calculated as 7! / (2! * (7-2)!). Therefore, the probability of two components failing is 0.2^2 * C(7, 2) * (0.8)^5 = 0.2605824.
Step 4: Find the probability of exactly three components failing.
Similarly, the probability of three components failing is (0.2)^3, and we can choose any three out of the seven components, so we need to multiply by the binomial coefficient (7 choose 3), denoted as C(7, 3), which is calculated as 7! / (3! * (7-3)!). Therefore, the probability of three components failing is 0.2^3 * C(7, 3) * (0.8)^4 = 0.1302912.
Step 5: Add up the probabilities from steps 1 to 4.
To find the probability of at most three components failing, we need to add up the probabilities from steps 1 to 4:
P(At most 3 components fail) = P(0 components fail) + P(1 component fails) + P(2 components fail) + P(3 components fail)
= 0.2097152 + 0.33554432 + 0.2605824 + 0.1302912
= 0.93613312.
Step 6: Find the probability that the machine will be working.
Since we're looking for the probability that the machine will be working, we can subtract the probability of at most three components failing from 1:
P(Machine working) = 1 - P(At most 3 components fail)
= 1 - 0.93613312
= 0.06386688.
Therefore, the probability that the machine will be working is approximately 0.0639, or 6.39%.