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Answer:
He expects that his bonus will be $210,000.
Step-by-step explanation:
For each plate appearence, there are only two possible outcomes. Either he gets on base(base hit or bases-on-balls), or he does not. The probability of getting on base on each plate appearence is independent of other plate appearences. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
The expected value of the binomial distribution is:
600 plate appearances this year
So n = 600.
Expected number of hits
So
Expected number of base on balls.
So
Bonus
$1,000 for each hit and $100 for each base-on-balls he gets.
200*1,000 + 100*100 = 210,000
He expects that his bonus will be $210,000.
Intersection of the first two lines:
Multiply the first equation by 4 and the second by 5:
Subtract the two equations:
Plug this value for y in one of the equation, for example the first:
So, the first point of intersection is
We can find the intersection of the other two lines in the same way: we start with
Use the fact that x and y are the same to rewrite the second equation as
And since x and y are the same, the second point is
So, we're looking for a line passing through and
. We may use the formula to find the equation of a line knowing two of its points, but in this case it is very clear that both points have the same coordinates, so the line must be
In the attached figure, line is light green, line
is dark green, and their intersection is point A.
Simiarly, line is red, line
is orange, and their intersection is B.
As you can see, the line connecting A and B is the red line itself.
