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# Beyond the Numbers: Delving into Calculus

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For this question, 60% is autonomous (constant) however the remaining 40% (that is, 100 – 60 = 40) depends on the ratio of time spent studying. This can be modeled by a linear equation of the form

Y = 60 + 40X

where; Y- average score and X- the ratio of time spent studying.

Given X = 0.5 (half) Y is obtained as.

Y = 60 + 0.5*40 = 80%.

Assuming I wanted a score of 95% then

95 = 60 +40X

X = 7/8 implying that I should study for at least 7/8 of the study time.

Question 2

In this question 30% is guaranteed, however, the remaining 50% (that is 80 – 30 =50) depends on the ratio of time spent studying. This can be modeled by a linear equation of the form

Y = 30 + 50X

where; Y- average score and X- the ratio of time spent studying.

Given X = 0.5 (half) Y is obtained as.

Y = 30 + 0.5*50 = 55%.

Assuming I wanted a score of 70% then

70 = 30 +50X

X = 4/5 implying that I should study for at least 4/5 of the time.

Question 3

Given probability of passing p = 0.6

Expected yards’ offense will gain from passing (E1) is obtained as follows:

E1

= 4*(0.6) + 9*(1-0.6)

= 2.4 + 3.6

= 6 yards

Expected yards’ offense will gain from run (E2) is obtained as follows:

E1

= 6*(0.6) + 1*(1-0.6)

= 3.6 + 0.4

= 4 yards

Therefore, the average yards the offense  can expect to gain is  = 5 yards.

Question 4

Given probability of passing p = 0.75

Expected yards’ defense will gain from passing (E3) is obtained as follows:

E3

= 2*(0.75) + 15*(1-0.75)

= 1.5 + 3.75

= 5.25 yards

Expected yards’ defense will gain from run (E4) is obtained as follows:

E4

= 5*(0.75) + 0*(1-0.75)

= 3.75 yards

Since the defense gain less yards for every run my next play will be a run

Question 5

#1 Confesses

# 1 Stays Silent

#2 Confesses

-5 years

-10 years

#2 Stays Silent

0 years

-6 months

The decision, in this case, will depend on the worst case scenario for the suspect. If #1 confess the worst can be 6 months in prison and the best can be 0 months. While if one stays silent the worst case can be 10 years and best 6 months. Therefore, suspect number one should confess.

Question 6

The odds of winning is 0.04 or 4 % (Multi-State Lottery Association). Therefore, expected payoff of a single lottery is 0.04*2 = \$0.08. When the expected payoff is at least half of the ticket price. That is 50% chance of winning.

Work Cited

Multi-State Lottery Association. Powerball. October 2015. HTML. 12 February 2018. <http://www.powerball.com/powerball/pb_howtoplay.asp>.

September 18, 2023
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Expertise Calculus
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