Question

Question 1: 11 ptsBased on results from recent track meets, Denita has a 68% chance of getting a medal in the 100 meter dash. Estimate the probability that Denita will get a medalin at least 5 of the next 10 races. Use the random number table, and make at least 10 trials for your simulation. Express your answer as a percent.

Answer 1

According to the binomial distribution of probability

$P(k;n,p)=P(X=k)=(nbinomialk)p^k(1-p)^{n-k}$

In our case,

$n=10,p=68\text{percent}=0.68$

Therefore,

$P(k\ge5)=\sum ^{10}_{k=5}(10binomialk)(0.68)^k(1-0.68)^{10-k}$

Thus,

$\Rightarrow P(k\ge5)=(10binomial5)(0.68)^5(0.32)^5+(10binomial6)(0.68)^6(0.32)^4+(10binomial7)(0.68)^7(0.32)^3+(10binomial8)(0.68)^8(0.32)^2+(10binomial9)(0.68)^9(0.32)^1+(10binomial10)(0.68)^{10}(0.32)^0$

After calculations, we reach the following result

$\Rightarrow P(k\ge5)=0.9362\ldots$

Then, the probability of Denita winning a medal in at least 5 of the next 10 races is, approximately 93.63%. The close option is 100%