CK-12-Basic Probability and Statistics Concepts - A Full Course

3.3. Binomial Distributions and Probability http://www.ck12.org

P(X=a) =nCa×pa×q(n−a)

P(X= 5 ) = 10 C 5 ×p^5 ×q(^10 −^5 )

P(X= 5 ) = 10 C 5 ×( 0. 306 )^5 ×( 0. 694 )(^10 −^5 )

P(X= 5 ) = 252 × 0. 00268 × 0. 161

P(X= 5 ) = 0. 109

Therefore, the probability of rolling at least one 2 exactly 5 times when 2 dice are rolled 10 times is 10.9%.

Practice

1. The probability of scoring above 80% on a math test is 20%. What is the probability of scoring below 80%?


2. The probability of getting a job after university is 85%. What is the probability of not getting a job after
   university?


3. A fair die is rolled 10 times. LetXbe the number of rolls in which we see a 6.
   a. What is the probability of seeing a 6 in any one of the rolls?
   b. What is the probability that we will see a 6 exactly once in the 10 rolls?


4. A fair die is rolled 15 times. LetXbe the number of rolls in which we see a 6.
   a. What is the probability of seeing a 6 in any one of the rolls?
   b. What is the probability that we will see a 6 exactly once in the 15 rolls?


5. A fair die is rolled 15 times. LetXbe the number of rolls in which we see a 5.
   a. What is the probability of seeing a 5 in any one of the rolls?
   b. What is the probability that we will see a 5 exactly 7 times in the 15 rolls?


6. A pair of fair dice is rolled 10 times. LetXbe the number of rolls in which we see at least one 5.
   a. What is the probability of seeing at least one 5 in any one roll of the pair of dice?
   b. What is the probability that in exactly half of the 10 rolls, we see at least one 5?


7. A pair of fair dice is rolled 15 times. LetXbe the number of rolls in which we see at least one 5.
   a. What is the probability of seeing at least one 5 in any one roll of the pair of dice?
   b. What is the probability that in exactly 8 of the 15 rolls, we see at least one 5?


8. A coin is flipped 25 times. LetXbe the number of flips in which we see a head.
   a. What is the probability of seeing a head in any one flip of the coin?
   b. What is the probability that we see a head in exactly 11 of the 25 flips?


9. A coin is flipped 30 times. LetXbe the number of flips in which we see a head.
   a. What is the probability of seeing a head in any one flip of the coin?
   b. What is the probability that we see a head in exactly 20 of the 30 flips?


10. A pair of coins is tossed 30 times. LetXbe the number of tosses in which we see at least one head.
    a. What is the probability of seeing at least one head in any one toss of the pair of coins?
    b. What is the probability that in exactly 25 of the 30 tosses, we see at least one head?