51.1k views
4 votes
A die is thrown 4 times. Getting a number greater than 2 is a success.

Find the probability of getting (i) exactly one success, (ii) less than 3 successes.

User Soli
by
7.8k points

1 Answer

1 vote

Final answer:

The probability of exactly one success is approximately 0.1317, and the probability of less than three successes is approximately 0.4395 when a die is rolled four times, with success being a roll greater than 2.

Step-by-step explanation:

Probability of Success on Die Rolls

When a die is thrown 4 times, and getting a number greater than 2 is considered a success, we can calculate the following probabilities:

(i) The probability of exactly one success can be determined by considering the success as rolling a 3, 4, 5, or 6, which has a probability of 4/6 (since there are four successful outcomes out of six total possible outcomes). The probability of failure (rolling a 1 or 2) is therefore 2/6.

Using the binomial probability formula, the probability of exactly one success in four trials (k=1 success, n=4 trials) is given by:

P(X=k) = (n choose k) p^k (1-p)^(n-k)

We find P(X=1) with p=4/6, q=2/6, n=4:

P(X=1) = (4 choose 1) (4/6)^1 (2/6)^(4-1) = 4 * (4/6) * (2/6)^3 = 0.1317 (rounded to four decimal places).

(ii) The probability of less than 3 successes includes the events of 0, 1, or 2 successes. We calculate each of these probabilities and add them together.

The sum of probabilities for 0, 1, or 2 successes is:

P(X<3) = P(X=0) + P(X=1) + P(X=2)

Calculating these individual probabilities with the binomial formula (and rounding to four decimal places) we get:

P(X=0) + P(X=1) + P(X=2) = 0.0444 + 0.1317 + 0.2634 = 0.4395

Therefore, the probability of getting less than 3 successes is 0.4395.

User Ryan Romanchuk
by
8.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories