Question

You are a surgical oncologist who specializes who specializes in removing liver tumors. You classify a surgery as a success if all tumor tissue is removed, and a failure if it is not. Each surgery has a probability of success p. You only perform one surgery per patient, and the surgeries are independent events, so you can consider each surgery as a Bernoulli trial.

1.  What is the probability of having at least 7 successes is you perform 10 surgeries with p = 0.75. Note that this is different than the question of having 7 successes if you perform 10 surgeries with p = 0.75. This question is asking for the probability that k ≥ 7, not just the probability that k = 7.

2. Suppose that you perform 10 different surgeries and p = 0.75. Make a graph with "number of successful surgeries" plotted on the horizontal axis and "probability" plotted on the vertical axis.

Answer 1

1.

Probability of having at least 7 successes = P( k ≥ 7) = P(k = 7) + P(k = 8) + P(k = 9) + P(k = 10)

= ¹⁰C₇ * 0.75⁷ * (1 - 0.75)³ + ¹⁰C₈ * 0.75⁸ * (1 - 0.75)² + ¹⁰C₉ * 0.75⁹ * (1 - 0.75)¹ + ¹⁰C₁₀ * 0.75¹⁰ * (1 - 0.75)⁰

= 120 * 0.75⁷ * 0.25³ + 45 * 0.75⁸ * 0.25² + 10 * 0.75⁹ * 0.25¹ + 1 * 0.75¹⁰ * 0.25⁰

= 0.7758751

2.

We use the below formula

P(X = k) =  ¹⁰C_k * 0.75^k * 0.25^10-k

to get the probability for each value of k.

For k = 1, .., 10, the probabilities for 10 different surgeries are,

0.0000286, 0.0003862, 0.0030899, 0.016222, 0.058399, 0.145998, 0.2502823, 0.2815676, 0.1877117, 0.0563135

The graph with "number of successful surgeries" plotted on the horizontal axis and "probability" plotted on the vertical axis is shown below.

R code used to generate the plot -

plot(c(1:10), dbinom(c(1:10),10,0.75), xlab = "number of successful surgeries" , ylab = "probability", pch = 19)
lines(c(1:10), dbinom(c(1:10),10,0.75), xlab = "number of successful surgeries" , ylab = "probability", pch = 19)