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: (a) To find the probability that at least 1 patient will survive for 10 years, we can calculate the probability of no one surviving and then subtract it from 1. (b) To find the probability that at least 10 patients survive, we have to add up the probabilities of having 10, 11, ..., 20 survivors using the binomial distribution formula. (c) Similarly, to find the probability that at least 15 patients survive, we have to add up the probabilities of having 15, 16, ..., 20 survivors using the binomial distribution formula.

Step-by-step solution

Find the probability of no one surviving 10 years

Since we want to find the probability of at least 1 person surviving, we can first find the probability of no one surviving and then subtract it from 1 (since it's a complementary event). We can use the binomial probability formula: \(P(x) = \binom{n}{x}p^{x}(1-p)^{n-x}\), where x is the number of successes, n is the number of trials, and p is the probability of success. In our case, x=0, n=20, and p=0.5. \(P(x=0) = \binom{20}{0}(0.5)^{0}(1-0.5)^{20-0}= (1)(1)(0.5)^{20}\).

Calculate the probability of at least 1 person surviving 10 years

Now that we know the probability of no one surviving, we can subtract it from 1 to find the probability of at least 1 person surviving. \(P(x \geq 1) = 1 - P(x=0) = 1 - (0.5)^{20}\). #b. At least 10 will survive for 10 years#

Find the probability using a cumulative formula

We need to find the probability that at least 10 people survive, which means 10, 11, 12, ..., 20 people could survive. We can do this by adding up the probabilities for each of these scenarios using the binomial probability formula. However, in practice, we would use a cumulative binomial probability table or software to calculate this. For the sake of this exercise, let's use the formula: \(P(x \geq 10) = \sum_{k=10}^{20} \binom{20}{k}(0.5)^{k}(0.5)^{20-k}\). #c. At least 15 will survive for 10 years#

Find the probability using a cumulative formula

We need to find the probability that at least 15 people survive, which means 15, 16, 17, ..., 20 people could survive. We can do this by adding up the probabilities for each of these scenarios, using the binomial probability formula. Again, in practice, we would use a cumulative binomial probability table or software to calculate this. For the sake of this exercise, let's use the formula: \(P(x \geq 15) = \sum_{k=15}^{20} \binom{20}{k}(0.5)^{k}(0.5)^{20-k}\).

Key concepts

Survival Rate

Survival rate is a key concept in medical research and statistics, often used to measure the effectiveness of treatments. It refers to the percentage of people who survive a certain disease for a specified period after diagnosis. In our exercise, the survival rate for bladder cancer over a 10-year period is 50%. This means if you have a group of 100 people diagnosed with bladder cancer, approximately 50 of them are expected to survive 10 years after treatment.
Understanding the survival rate can help doctors and patients gauge the overall risk and make informed decisions about treatment options and prognosis.
It's important to remember that individual outcomes can vary. The survival rate gives a general overview but not a guarantee for every individual case.

Complementary Events

In probability, complementary events are pairs of events where one must happen if the other does not. They combine to cover all possible outcomes. For example, in our bladder cancer scenario, if we know the probability of someone not surviving 10 years (the event's complement), we can easily find the probability of survival by using complementary events.
The probability of at least one person surviving in a group of 20 is the complement of no one surviving. Mathematically, if the probability of no one surviving is calculated as \(0.5^{20}\), the probability of at least one person surviving is calculated as \((1 - 0.5^{20})\).
Using the concept of complementary events helps simplify complex probability calculations by working with what is often easier to calculate.

Cumulative Probability

Cumulative probability refers to the probability of obtaining a value less than or equal to a given point in a probability distribution. In terms of binomial probability, it's the likelihood that a random variable falls within a particular range.
In our case, we're looking at the cumulative probability of having at least a certain number of survivors out of 20 people. For instance, to find the probability that at least 10 people survive, we sum the probabilities of exactly 10, 11, 12, ..., up to 20 survivors. This is noted as \((P(x \geq 10) = \sum_{k=10}^{20} \binom{20}{k}(0.5)^{k}(0.5)^{20-k})\).
The cumulative approach is useful because it saves time and calculation effort, especially when seeking the total probability of multiple specific outcomes.

Probability of Success

The probability of success in a binomial distribution is the chance that a single trial results in the desired outcome. Here, the probability of success is considered each patient's 10-year survival, estimated at 50% or 0.5.
This probability is crucial when using the binomial probability formula, which offers a way to calculate the likelihood of achieving various numbers of successes (like survivors) over several trials (or patients).
Understanding the "probability of success" allows us to use mathematical formulas to predict various survival scenarios and make informed decisions based on those predictions. In many scenarios, knowing this probability aids in planning and evaluating potential outcomes.