Question

Consider a binomial random variable with \(n=9\) and \(p=.3 .\) Let \(x\) be the number of successes in the sample. Evaluate the probabilities in Exercises \(7-10 .\) $$ P(x>2) $$

Short answer

Answer: To find the probability that the number of successes is greater than 2, we first need to calculate the complement, which is the probability that the number of successes is less than or equal to 2. Using the binomial probability formula and the complement rule, we find the probability of the number of successes being greater than 2. After calculating the values, the probability of having more than 2 successes in the 9 trials is given by: P(x>2) = 1 - P(x≤2) P(x>2) = 1 - (P(x=0) + P(x=1) + P(x=2)) Compute the exact probabilities for P(x=0), P(x=1), and P(x=2), then substitute them into the equation and calculate the result.

Step-by-step solution

Understand the binomial probability formula

The binomial probability formula is given by: $$ P(x=k) = {n \choose k} p^k (1-p)^{n-k}, $$ where \(n\) is the number of trials, \(p\) is the probability of success, and \(k\) is the number of successes we are interested in.

Recall the complement rule

The complement of an event \(A\) is denoted by \(A'\) and is the event that \(A\) does not occur. To use the complement rule, we recall the fact that the probability of an event occurring is equal to \(1\) minus the probability of its complement. Mathematically, we can write this as: $$ P(A) = 1 - P(A'). $$

Apply the complement rule to the given problem

We are asked to find the probability \(P(x>2)\). To apply the complement rule, we first find the complement of the event, which is \(x \le 2\). Thus, we can write the desired probability as: $$ P(x>2) = 1 - P(x\le 2). $$

Calculate the probability using the binomial formula

To calculate \(P(x\le 2)\), we need to determine the probabilities for each value of \(x\) in the range \(0\) to \(2\). In other words, we need to calculate \(P(x=0)\), \(P(x=1)\), and \(P(x=2)\). Using the binomial probability formula, we have: $$ P(x=0) = {9 \choose 0} (0.3)^0 (1-0.3)^{9-0} \\ P(x=1) = {9 \choose 1} (0.3)^1 (1-0.3)^{9-1} \\ P(x=2) = {9 \choose 2} (0.3)^2 (1-0.3)^{9-2} $$

Sum the probabilities and find the complement

Now we sum the probabilities \(P(x=0)\), \(P(x=1)\), and \(P(x=2)\) to find \(P(x\le 2)\): $$ P(x\le 2) = P(x=0) + P(x=1) + P(x=2). $$ Finally, we substitute this into our complement rule equation to find \(P(x>2)\): $$ P(x>2) = 1 - P(x\le 2). $$ Calculate the exact values for each probability and you will have the desired result for \(P(x>2)\).

Key concepts

Binomial Random Variable

A binomial random variable represents the number of successes in a set number of independent trials in a binomial experiment. To qualify as a binomial experiment, several conditions must be met:

• There must be a fixed number of trials (denoted by n).
• Each trial can result in just two possible outcomes: success or failure.
• The probability of success (p) must be constant for each trial.
• Trials must be independent, meaning the outcome of one trial doesn’t affect another.

In the given exercise, we have a binomial random variable with 9 trials (in other words, n = 9) and the probability of success for each trial is 0.3 (so, p = 0.3). You are asked to calculate the probability of getting more than 2 successes (x > 2). Understanding what a binomial random variable is, helps in identifying the appropriate method to calculate probability and using the correct probability formula.

Probability of Success

The probability of success in the context of binomial distribution is the likelihood of a successful outcome occurring during a single trial. In our example, a 'success' could be any desired outcome we're examining in the experiment, and every trial has the same chance or probability of success, which is symbolized as p. Remember, the complement of the probability of success is the probability of failure, which occurs with probability 1 - p.

When calculating binomial probabilities, it's crucial to know this probability, as our calculations will heavily rely on it to determine the likelihood of seeing a particular number of successful outcomes across the trials. For instance, if p = 0.3, then for any given trial, the event has a 30% chance of happening (success), while the probability of it not happening (failure) would be 70% (1 - 0.3 = 0.7). This insight is the foundation for computing probabilities using the binomial probability formula.

Complement Rule

The complement rule is a fundamental concept in probability theory used to find the probability of the occurrence of the opposite of a particular event, referred to as its complement. According to this rule, the sum of the probabilities of an event and its complement is always 1. This is expressed mathematically as P(A) + P(A') = 1, where P(A') denotes the probability of the complement of A.

By rearranging the equation, we can derive the useful form P(A) = 1 - P(A'). This allows us to calculate the probability of an event by knowing the probability of its complement, which is particularly handy in cases where it's easier to calculate the probability of the complement rather than the event itself. In the example of our exercise, we used the complement rule to find P(x > 2) by first calculating P(x ≤ 2) and then subtracting from 1, simplifying our work by avoiding the direct calculation of P(x > 2) which would require more extensive computation.