hospital records show that patients suffering from a specific disease die sixty percent of the time what is the probability that of five patients suffering from this disease three will survive this is an example of a by normal distribution where a by normal distribution is a discrete probability distribution in which one the experiment consists of n independent trials to each trial has two outcomes success or failure and three the probability of success is little P which remains constant for each trial for a by normal distribution the probability of having our successes is equal to n choose R times P raised to the power of r times the quantity 1 minus P raised to the power of n minus R for our situation P will represent the probability of survival and because a patient will die 60% of the time little P the probability of survival is equal to 1 minus 60% which is equal to 40% or as a decimal 0.4 and because we are selecting 5 patients and the number of independent trials is 5 and because we are trying to determine the probability that three patients will survive are the number of successes is equal to 3 and therefore the probability that 3 of 5 survive is the probability that X is equal to 3 which is equal to n choose R which is 5 choose 3 times P raised to the power of R which is 0.4 the probability of survival raised a power of 3 the number of successes times the quantity 1 minus P which is 1 minus 0.4 0.6 which is the probability of death raised to the power of n minus R which in our case is 5 minus 3 which is equal to 2 once we have this setup notice how the sum of the exponents will always be equal to n and now to evaluate this 5 to 3 equal to five factorial divided by the product of our factorial and n minus R factorial where R factorial is 3 factorial and n minus R factorial is 2 factorial that's going to evaluate this quotient 5 factorial is equal to 5 times 4 times 3 times 2 times 1 which is 3 factorial so let's write this as 5 times 4 times 3 factorial divided by the product of 3 factorial and 2 factorial and then we have times 0.4 to the third time's a 0.6 squared simplifying here 3 factorial divided by itself simplifies to 1 5 times 4 is 20 to 40 times 1 or 2 20 divided by 2 is equal to 10 giving us 10 times 0.4 to the third time's the 0.6 squared and now we go the calculator 10 times the 0.4 cubed times 0.6 squared is equal to 0.2 3 0 4 which is a percentage is 23.0 4% the probability that other five patients suffering from this disease three will survive is twenty three point zero four percent I hope you found this helpful