Question - Understanding the Binomial Distribution in Probability Theory
Solution:
The image contains a question in French about probability. The question translates to:"We know from experience that a certain surgical operation has a 95% chance of success. We are about to perform this operation on 6 patients. Let X be the random variable equal to the number of successful operations out of the 6 attempts.1) What is the law followed by X?"The law followed by X is the binomial distribution. This is because the binomial distribution is applicable for a fixed number of independent trials (in this case, 6), where each trial has only two possible outcomes (success or failure), and the probability of success is the same for each trial.In mathematical terms, if $$ p $$ is the probability of success for each trial, $$ n $$ is the number of trials, and $$ k $$ is the number of successes, the probability $$ P(X = k) $$ is given by:\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]For this particular case:- $$ n = 6 $$ (the number of patients),- $$ p = 0.95 $$ (the probability of success for the surgical operation),- $$ k $$ can be any integer from 0 to 6 (the number of successful operations).So the random variable $$ X $$ representing the number of successful operations follows a binomial distribution with parameters $$ n = 6 $$ and $$ p = 0.95 $$.
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