Answer:
To reach her goal in the first 100 serves she has to have 50 successful services. This happens with a probability of 2%.
That means she has 98% chances of needing more than 100 serves to reach her goal.
Step-by-step explanation:
The condition for Jo to need more than 100 serves is that she do not accomplish 50 successful serves in 100 tries.
We have a probability sample of possible results that hass a binomial distribution, with n=100 and p=0.4.
As n is large enough, we will approximate it with a normal distribution. The parameters of the normal distribution are:
[tex]\mu=np=100\cdot0.4=40\\\\\sigma=\sqrt{np(1-p)}=\sqrt{100\cdot0.4\cdot0.6}=\sqrt{24}=4.9[/tex]
We need the to calculate the probability that the amount of successful services is less than 50, so that she needs more than 100 serves. This is:
[tex]P(X<50)[/tex]
We calculate the z value for X:
[tex]z=(X-\mu)/\sigma=(50-40)/4.9=2.04[/tex]
Then we have
[tex]P(X<50)=P(z<2.04)=0.98[/tex]
That means that there is 98% probability that she will need more than 100 serves to reach her goal.
