S5 A bag contains twenty counters ,numbered 1.2、 A counter is selected at random from the bag its number IS noted , and it is then returned to the ba (i)If this operation is carried out 100 times find, 2 significant figures , the probability that the num selected at least A times. (ii)If the operation is carried out 3 times find th probability that the largest number drawn is n . W 1leqslant nleqslant 20 . Find the expected value of n to 3 sig figures.

## Step 1: Identify the total number of counters and the probability of selecting any one counter.There are 20 counters, each numbered from 1 to 20. The probability of selecting any one counter is: ## Step 2: Calculate the probability of selecting a counter at least 4 times in 100 trials.We use the binomial distribution formula \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\), where is the number of trials, is the number of successes, and is the probability of success in a single trial.Here, , , and .The probability of selecting a counter at least 4 times is: Calculate each term: Sum these probabilities and subtract from 1 to get \(P(X \geq 4)\).## Step 3: Calculate the probability of the largest number drawn being in 3 trials.The probability of the largest number drawn being is given by: For trials, the probability that all numbers are is: And for : Thus: ## Step 4: Calculate the expected value of .The expected value \(E(n)\) is given by: Substitute the probabilities calculated in Step 3 and sum over all from 1 to 20.#