In a binomial experiment, the probability of success in each trial is 0.45 and 30 trials are performed. Let 𝑋 be the random variable, which counts the number of successes. Find, to two decimal places, the standard deviation of 𝑋.
Let’s begin by looking at the two key properties of a binomial experiment. Firstly, it consists of 𝑛 independent repeated trials. Each of these trials has two possible outcomes: success and failure. This means that there are two key values when dealing with a binomial experiment, denoted by 𝑛 and 𝑝. 𝑛 is the number of trials and 𝑝 is the probability of success.
In this particular question, there are 30 trials and the probability of success is 0.45. The expected value or mean of any binomial experiment is denoted by 𝐸 of 𝑋. This is equal to 𝑛 multiplied by 𝑝. The variance, or Var of 𝑋, is equal to 𝑛 multiplied by 𝑝 multiplied by one minus 𝑝. The standard deviation, which we need to calculate in this question, is equal to the square root of the variance.
We begin by calculating the variance. This is equal to 30 multiplied by 0.45 multiplied by 0.55. Typing this into our calculator gives us 7.425. The standard deviation is the square root of this value. The square root of 7.425 is equal to 2.724885 and so on. As we need to round our answer to two decimal places, the four is the deciding value. As this is less than five, we will round down.
The standard deviation of a binomial experiment with 30 trials and probability of success of 0.45 is 2.72.