# Baseball star David Ortiz-nicknamed "Big Papi"-is known for his ability to deliver hits in high-pressure situations. Here is a two-way table of his hi

Baseball star David Ortiz-nicknamed "Big Papi"-is known for his ability to deliver hits in high-pressure situations. Here is a two-way table of his hits, walks, and outs in all of his regular-season and post-season plate appearances from 1997 through 2014. Choose a plate appearance at random. Are the events "Hit" and "Post-season" independent? Justify your answer.
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DEFINITIONS Two events are independent, if the probability that one event occurs in no way affects the probability of the other event occurring. Definition conditional probability: $P\left(A\mid B\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{P\left(A\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}B\right)}{P\left(A\right)}$ SOLUTION We note that the table contains information about 8883 at-bats (given in the bottom right corner of the table). Moreover, 2110 of the 8883 at-bats are Hits, because 2110 is mentioned in the row ” Total” and in the column ” Hit” of the table. The probability is the number of favorable outcomes divided by the number of possible outcomes:
Next, we note that 352 of the 8883 at-bats are post-season, because 352 is mentioned in the row ”Post” and in the column ”Total” of the given table.
Next, we note that 87 of the 8883 at-bats are hits in post-season, because 87 is mentioned in the row ”Post” and in the column ”Hit” of the given table. Use the definition of conditional probability: $P\left(Hit\mid Post\right)=\frac{P\left(Hit\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Post\right)}{P\left(Post\right)}$
$\frac{\frac{87}{8883}}{\frac{352}{8883}}$
$\frac{87}{352}$
$\approx 0.2472$
$=24.72\mathrm{%}$
If events A and B are independent, then $P\left(A|B\right)=P\left(A\right)$ and $P\left(B|A\right)=P\left(B\right)$
In this case, we note that $P\left(Hit|Post\right)=0.2472$ is not the same as $P\left(Hit\right)=0.2375$ and thus the two events are not independent.