Question

4.The success rate of corneal transplant surgery is 85%. The surgery is performed on six patients.

(a) Is this a binomial distribution? (hint: check conditions!)
This is A Bihmich Distribution
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(b) Make a probability dist. table
(c) Graph the distribution using a histogram and describe it (SCUFS)

Answer 1

The corneal transplant surgery, with an 85% success rate, follows a binomial distribution (B(6, 0.85)). The histogram of this distribution is likely right-skewed. The mean is 5.1 surgeries, and the variance (0.765) and standard deviation (0.875) provide insights into the variability around the expected success rate.

a. Binomial Distribution:

In a binomial distribution, each trial is independent, and there are only two possible outcomes: success or failure. The probability of success remains constant across all trials. In this case, the corneal transplant surgery has a success rate of 85%, and it is performed on six patients.

The binomial distribution can be expressed as B(n, p), where n is the number of trials and p is the probability of success.

b. Graphing the Binomial Distribution:

Using a histogram, each bar represents the probability of a specific number of successful surgeries in the six trials. The shape of the distribution is likely to be skewed to the right, with the peak around the mean.

c. Mean, Variance, and Standard Deviation:

The mean (
`\(\mu\)`) of a binomial distribution is calculated as np, where n is the number of trials and
`\(p\)` is the probability of success. For this scenario,
`\(\mu = 6 * 0.85 = 5.1\).`

The variance (
`\(\sigma^2\)`) is
`\(np(1-p)\)`. Here,
`\(\sigma^2 = 6 * 0.85 * 0.15 = 0.765\).`

The standard deviation (
`\(\sigma\)`) is the square root of the variance. Thus,
`\(\sigma \approx √(0.765) \approx 0.875\).`

Interpretation: The mean (5.1) represents the expected number of successful surgeries, while the standard deviation (0.875) indicates the spread or variability around the mean. The variance quantifies the overall variability in the number of successful surgeries.

The probable question may be:

The success rate of corneal transplant surgery is 85%. The surgery is performed on six patients.

a. Construct a binomial distribution.

b. Graph the binomial distribution using a histogram and describe its shape.

c. Find the mean, variance, and standard deviation of the binomial distribution and interpret the results.