Brandon rolls 1 d4 20 times, counting the number of times \'4\' comes up. (a) Ho
ID: 3132036 • Letter: B
Question
Brandon rolls 1 d4 20 times, counting the number of times '4' comes up. (a) How is "success" defined for this set-up? (b) How is "failure" defined? (c) How many ways are there to get exactly five '4's? (d) Find the probability Brandon gets exactly five '4's. 3. Which of these problems could be figured out using the binomial distribution? Circle the letter for any that work. (a) Figuring out the probability of exactly 8 heads in 20 coin tosses. (b) Figuring out the probability of being dealt a royal flush in poker. (c) Finding the probability of 4 or more sixes when rolling a fair die 15 times in a row. (d) Finding the probability that the fourth roll of a die yields the first 6.Explanation / Answer
brandon rolls one 4 sided die numbering 1,2,3,4 20 times and counting the number of times '4' comes up
a) hence here "success" is "getting a 4 in one roll of 1d4"
b) here failure is termed as "not getting a 4 in one roll of 1d4"
c) there are all total 20 throws. out of which 5 '4's can be obtained from any 5 throws of the 20 throws in 20C5=15504 ways
d) let X be the random variable denoting the number of times '4' comes up out of 20 throws.
now probability that a 4 comes up in 1d4 is 1/4
hence X~Bin(20,1/4)
so pmf of X is P[X=x]=20Cx(1/4)x(1-1/4)20-x x=0,1,2,3,.....,20
hence the probability that brandon gets exactly 5 '4's is
P[X=5]=20C5(1/4)5(1-1/4)20-5=15504(0.25)5(0.75)15=0.202331 [answer]
3. (a) can be figured by a binomial distribution
if X be the number of heads appear in 20 coin tosses then X~Bin(20,0.5)
and we need to find P[X=8]
(c) can be figured by a binomial distribution
if Y denotes the number of sixes in rolling a fair die 15 times then Y~Bin(15,1/6)
and we need to find P[Y>=4]