Question

The 10 -year survival rate for bladder cancer is approximately \(50 \%\). If 20 people who have bladder cancer are properly treated for the disease, what is the probability that: a. At least 1 will survive for 10 years? b. At least 10 will survive for 10 years? c. At least 15 will survive for 10 years?

Short answer

Answer: Use the binomial probability formula to find the probabilities of at least 1, 10, and 15 people surviving for 10 years. Calculate the complementary probabilities for at least 1 survivor (1 - P(X=0)) and sum the individual probabilities for at least 10 and 15 survivors using the binomial probability formula.

Step-by-step solution

Definition of Binomial probability formula

Given n independent trials with a probability p of success, the binomial probability formula calculates the probability of exactly k successes as: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\) Where \(P(X=k)\) represents the probability of exactly k successes, \(\binom{n}{k}\) represents the binomial coefficient (the number of ways to choose k successes from n trials), p represents the probability of success, and (1-p) represents the probability of failure. In this exercise, n=20 (number of people with bladder cancer), p=0.5 (probability of a person surviving for 10 years), and k is the number of survivors. We will apply this formula to find the probabilities in each case.

a. At least 1 will survive for 10 years

To find the probability that at least 1 of the 20 people will survive for 10 years, we can use the complement rule, which states that the probability of an event occurring is equal to 1 minus the probability of that event not occurring. In this case, the event we want to consider is the complementary event of none of the 20 people surviving, and we want to find its probability. Using the binomial probability formula, we can find the probability of no one surviving for 10 years as: \(P(X=0) = \binom{20}{0} 0.5^0 (1-0.5)^{20-0} \) Now we can calculate the probability of at least 1 person surviving as: \(P(X ≥ 1) = 1 - P(X=0)\)

b. At least 10 will survive for 10 years

To find the probability that at least 10 of the 20 people will survive for 10 years, we have to find the probability of 10, 11, ..., 20 survivors and add them together. We can do this using the binomial probability formula: \(P(X ≥ 10) = P(X=10) + P(X=11) + ... + P(X=20)\)

c. At least 15 will survive for 10 years

Similar to the case with at least 10 survivors, for at least 15 survivors, we have to find the probability of 15, 16, ..., 20 survivors and add them together using the binomial probability formula: \(P(X ≥ 15) = P(X=15) + P(X=16) + ... + P(X=20)\) By calculating the respective probabilities using the binomial formula, we can find the probabilities for each of the cases.

Key concepts

Probability of Survival

Understanding the probability of survival involves determining how likely it is for an individual or a group to continue living or existing under certain conditions over a specified period. In the context of health and medical studies, such as assessing the effectiveness of treatments for diseases like bladder cancer, the probability of survival is a critical metric. It helps in estimating the percentage of individuals expected to survive following a diagnosis or treatment.

For instance, if the 10-year survival rate for bladder cancer is approximately 50%, we interpret this as there being a 50% chance for each individual in a group to survive for at least 10 years after being properly treated for the disease. To calculate the probability that a specific number of people within a group will survive, we use the concepts of binomial probability and statistical methods.

Probability Distribution

A probability distribution describes how the probabilities of various possible outcomes are distributed in a random process. It outlines the likelihood of different results happening and can take various forms like binomial, normal, or Poisson distribution, among others.

The binomial probability distribution, for instance, applies to scenarios with a fixed number of independent trials, each with two possible outcomes: success or failure. In our survival rate example, surviving for 10 years is considered a 'success', while not surviving is a 'failure'. The distribution would thus showcase the probability of having 0, 1, 2, ..., up to 20 survivors in a group of 20 individuals, given the 50% survival rate. It's crucial to comprehend the attributes of these distributions to predict outcomes effectively.

Complement Rule

The complement rule is a fundamental principle in probability that facilitates the calculation of the likelihood of an event by using the probability of its complement. An event's complement is essentially what happens when the event does not occur.

In mathematical terms, if the probability of an event A occurring is represented by P(A), then the probability of A not occurring, which is the complement of A, denoted as P(A'), is calculated as 1 - P(A). The rule simplifies calculations, especially when it's easier to find the probability of the complement. For example, calculating the probability that at least one person survives is much simpler by first determining the chance that no one survives and then subtracting it from 1.

Binomial Coefficient

The binomial coefficient is a key part of the binomial probability formula and represents the number of ways to choose a given number of successes from a set number of trials. It's denoted by \( \binom{n}{k} \) and is calculated according to the formula \( \frac{n!}{k!(n-k)!} \) where 'n' is the total number of trials, 'k' is the number of successes, and '!' represents the factorial operation.

In our exercise, the binomial coefficient comes into play when determining how many unique groups of survivors could exist. For example, out of 20 people, the binomial coefficient tells us how many ways we can choose 1 survivor, 10 survivors, or 15 survivors. This coefficient is crucial for accurately applying the binomial probability formula to find the probabilities of the various survival scenarios presented in the exercise.