Question

15 multiple choice 2 points
a certain baseball player has a batting average (number of hits/number of times at bat) of 0.416. what is the probability that he will not get a hit in his next time at bat?
p(not a hit) = 0.416
p(not a hit) = 1.416
p(not a hit) = 0.574
p(not a hit) = 0.584
16 multiple choice 2 points
determine the combination $_{14}c_{10}$.
576,576
0.0000002756
1,001
3,632,428,800

Explanation:

Question 15

Step1: Understand the problem

The batting average is the probability of getting a hit, \( P(\text{hit}) = 0.416 \). The probability of an event and its complement (not happening) sums to 1. So, \( P(\text{not a hit}) = 1 - P(\text{hit}) \).

Step2: Calculate the probability

\( P(\text{not a hit}) = 1 - 0.416 = 0.584 \)

Step1: Recall the combination formula

The combination formula is \( _nC_r=\frac{n!}{r!(n - r)!} \). For \( _{14}C_{10} \), we can also use the property \( _nC_r = _nC_{n - r} \), so \( _{14}C_{10}=_{14}C_{4} \) (since \( 14 - 10 = 4 \)).

Step2: Calculate \( _{14}C_{4} \)

\( _{14}C_{4}=\frac{14!}{4!(14 - 4)!}=\frac{14!}{4!×10!}=\frac{14×13×12×11}{4×3×2×1} \)
\( \frac{14×13×12×11}{24}=\frac{24024}{24} = 1001 \)

Answer:

\( P(\text{not a hit}) = 0.584 \) (the option with this value)

Question 16