Question

When Jerome plays darts. The chances they hit the bulls eye is 1/2.

a)Draw a Tree diagram to represent the problem when three darts are thrown.

b)What is the probability three darts in a row hit the bullseye?

c)What is the probability that none of the three hit the bullseye?

d)What is the probability at least one of the three will hit the bullseye?

e)What is the probability exactly 2 of the three hit the bullseye?

Answer 1

Let H denote the hit and M denotes the miss of the dart.

The outcomes of the experiment are {HHH,HMM,MHM,MMH,HHM,HMH,MHH,MMM}

(b)What is the probability three darts in a row hit the bullseye?

P(three darts in a row hit the bullseye)=P(first dart hit the bullseye)P(second dart hit the bullseye)P(third dart hit the bullseye)=P((HHH))=

(c)What is the probability that none of the three hit the bullseye?

P(none of the tree hit the bullseye)=P(first dart not hit the bullseye)P(second dart not hit the bullseye)P(third dart hit the bullseye)=P(MMM)=

(d)What is the probability at least one of the three will hit the bullseye?

Let H denote the hit and M denotes the miss of the dart.

P(HHH)= (Probability that All three darts hits the bulls eye)

P(MHH)= (First dart misses and the second and last hits the bulls eye)

P(HMH)=

P(HHM)=

P(MMH)=

P(MHM)=

P(HMM)=

Therefore, Probability of hitting the target atleast one is the sum of the above probabilities=.

(e)What is the probability exactly 2 of the three hit the bullseye?

Exactly two hits in three darts can be occured in the following ways, (HHM),(HMH),(MHH)

Hence the probability of exactly 2 of the three hit the bullseye=P( (HHM),(HMH),(MHH))=