The probability that a patient in a certain provincial...

The probability that a patient in a certain provincial hospital recovers from a heart disease is $35 \%$. In a sample of 12 patients with heart disease, find the probability:
a. that at least 10 will survive
b. that 5 to 11 will survive
c. that exactly 10 will survive

Key Concepts

Cumulative Probabilities

Cumulative probabilities involve summing individual probabilities to find the probability that a random variable falls within a specific range, either up to a given value or between two values. This concept is important when the problem requires calculating the probability of achieving at least or at most a certain number of successes.

Combinatorial Analysis

Combinatorial analysis is the branch of mathematics dealing with combinations and permutations, which is fundamental for determining the number of ways events can occur. In binomial probability problems, it is used to compute the binomial coefficients, which represent the number of ways to choose a specific number of successes from a set of trials.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood of a given number of successes in a fixed number of independent trials, with each trial having the same probability of success. It is particularly useful in scenarios where there are two possible outcomes, such as success or failure, across a set number of trials.

Probability Mass Function (PMF)

The probability mass function is a function that provides the probability that a discrete random variable is exactly equal to a particular value. In the context of the binomial distribution, the PMF calculates the probability of obtaining exactly k successes out of n trials using the binomial coefficient and the probabilities of success and failure.

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