can someone help me with question 2. theres no worked examples in my book for it.
this is methods units 3 & 4
2. Sammy and Guido play racquetball each week. In any one game, Sammy has the probability of
winning of 0.7. The outcome of any one game between the two is independent of the outcome of the
previous games played.
a. If the play 8 games, what is the probability, to four decimal places, that Sammy wins 5 out of 8
games?
b. What is the probability, to four decimal places, that Guido wins at most 6 of the 8 games?
c. What is the probability, to four decimal places, that Sammy only wins the second, fourth, fifth,
seventh and eighth games?
d. Compare the value obtained in part (a) with that of part (c). Explain the result.
e. How many games do they need to play so that Guido has at east 80% chance of winning at least
one game?
let 'S' be the random variable 'Sammy wins a game' and let 'G' be the random variable 'Guido' wins a game
a. S~Binomial(n=8,p=0.7) i.e. S follows a binomial distribution where the number of trials is 8 and the probability of success is 0.7
Solve Pr(S=5) using the cas or by hand using the binomial formula
b. G~Binomial(n=8,p=0.3) i.e. G follows a binomial distribution where the number of trials is 8 and the probability of success is 1-0.7=0.3
Solve Pr(G ≤ 6) with the cas or you can do it by hand by using the binomial formula and adding up Pr(G=1) + Pr(G=2) ... + Pr(G=6)
c. Think you can use a tree diagram for this one so
let S=sammy wins and G=guido wins
G*S*G*S*S*G*S*S
d. (c) would be lower than (a) due to the order Sammy wins or loses mattering in part (c) compared to (a) where the order does not matter i.e. Sammy can still get 5 wins without winning the second, fourth, fifth, sixth or eighth game resulting in a different combination.
e. Pr(G≥1)≥0.8
G~Binomial(n=?, p=0.3)
put in random trial values into the binomial cdf function until you find a value close to 0.8 as it says at least 80% but not below it
binomCdf(n=8,p=0.3,lower bound=1, upper bound=n=8) = 0.94 so too high
binomCdf(n=5,p=0.3,lower bound=1, upper bound=n=5) = 0.83
binomCdf(n=4,p=0.3,lower bound=1, upper bound=n=4) = 0.76 so too low
So the minimum number of games Guido needs to play to have at least 80% of winning is 5
Not sure if these are all correct so someone please correct me if they're wrong