Next Step PCAT Public Office Hours: Probability and Calculus · 2019-11-13 · Suppose we randomly...

Office Hours: Biology Content Review & Difficult Passage

Next Step PCAT Public Office Hours:

Probability and Calculus

The

Pharmacist

Is In

• Welcome & Introduction!

• Warm Up

• Probability Practice

• Calculus Practice

Introduction to Office Hours

• These sessions are meant to be:

• Thanks for coming to Next Step PCAT Office Hours!

• Today’s focus: Quantitative Reasoning

Interactive

Problem-focused

Specific to your needs (so ask questions!)

• This is not just a lecture! You can benefit most by:

Raising your hand and speaking

Commenting in the Question/Chat box

Participating!

Before Getting Started

1. If you have a microphone, make sure

it is turned on and easily available.

2. Locate the hand-raise button on the

toolbar on your screen.

3. Locate the Q box on the toolbar.

4. Let me know if you’re having any

technical issues!

• If you haven’t been here before, here’s how it works…

Office Hours: Open Q&A

If on wireless connection:

• Close any other internet resource-heavy processes

• Ask other users on network to do the same

• Sit as close to router as possible

Introduction

Hi, I’m Sophia!

• PCAT Content Manager

• 99th percentile PCAT

• 7 years teaching experience

• In my free time, I love running and working on jigsaw puzzles.

We will have some great free resources available for you at the end of the

presentation. Stick around and give us some feedback on our short survey as well.

Warm UpSuppose we randomly select two cards out of a 52-card deck. What is the probability that both are queens if we replace the first card? What is the probability if we do not replace the first card?

P first queen =4

52=

1

13

P second queen =1

13

P two queens =1

13×

1

13=

1

169

P first queen =4

52=

1

13

P second queen =3

51=

1

17

P two queens =1

13×

1

17=

1

221

Warm UpFind the minima or maxima of the function f(x) = 3x2 – 2x +4.

𝑓 x = 3x2 − 2x + 4

𝑓′ x = 6x − 2

0 = 6x − 2

2 = 6x

x =1

3

𝑓1

3= 3

1

3

2

− 21

3+ 4 =

1

3−2

3+12

3=11

3

𝑓′ x = 6x − 2

𝑓′′ x = 6

Minimum at1

3,11

3

Working with Probability EquationsLet’s say we have 10 blue marbles, 6 red marbles, and 2 green marbles. What is the probability of drawing…

1. one blue marble?

𝐏 𝐛𝐥𝐮𝐞 =𝟏𝟎

𝟏𝟎 + 𝟔 + 𝟐=𝟏𝟎

𝟏𝟖=𝟓

𝟗


2. NOT a blue marble?

𝐏 𝐧𝐨𝐭 𝐛𝐥𝐮𝐞 = 𝟏 −𝟓

𝟗=𝟗

𝟗−𝟓

𝟗=𝟒

𝟗


3. a blue OR a red marble?

𝐏 𝐛𝐥𝐮𝐞 𝒐𝒓 𝐫𝐞𝐝 = 𝐏 𝐛𝐥𝐮𝐞 + 𝐏 𝐫𝐞𝐝

=𝟓

𝟗+

𝟔

𝟏𝟎 + 𝟔 + 𝟐=𝟓

𝟗+

𝟔

𝟏𝟖=𝟓

𝟗+𝟑

𝟗=𝟖

𝟗


4. one blue marble AND one red marble (w/ replacement)?

𝐏 𝐛𝐥𝐮𝐞 𝒂𝒏𝒅 𝐫𝐞𝐝 = 𝐏 𝐛𝐥𝐮𝐞 × 𝐏 𝐫𝐞𝐝

=𝟓

𝟗×𝟏

𝟑=

𝟓

𝟐𝟕


5. one blue marble AND then one red marble (w/o replacement)?

𝐏 𝐛𝐥𝐮𝐞 𝒂𝒏𝒅 𝐫𝐞𝐝 = 𝐏 𝐛𝐥𝐮𝐞 × 𝐏 𝐫𝐞𝐝|𝐛𝐥𝐮𝐞

=𝟓

𝟗×

𝟔

𝟗 + 𝟔 + 𝟐=𝟓

𝟗×

𝟔

𝟏𝟕=𝟓

𝟑×

𝟐

𝟏𝟕=𝟏𝟎

𝟓𝟏

Working with Probability EquationsPick a number between 1 and 20. What is the probability that we select a number that is divisible AT LEAST by 3 or 4?

Numbers divisible by 3:Numbers divisible by 4:

Add these up:

BUT we double-counted 12!

P(at least A or B) = P(A) + P(B) – P(A and B)P(div. by 3 or 4) = P(div. by 3 or 4) =

3, 6, 9, 12, 15, 184, 8, 12, 16, 20

11/20

+ [4, 8, 12, 16, 20][3, 6, 9, 12, 15, 18] – [12](6/20) + (5/20) – (1/20) = 10/20 = 1/2

Working with Probability EquationsCan you identify which probability equation to use?

1. I have 5 different shirts, 2 different pants, and 4 different pairs of shoes. What is the probability that I’m wearing my green shirt with brown pants and black shoes?

2. How many different ways can the letters in the word CANDLE be rearranged?

3. There are 10 girls and 9 boys in the 3rd grade class. What is the probability that 3 students randomly chosen to read from their textbook will all be boys?

4. The volleyball coach must select 6 players to start the match out of 12 members of the volleyball team. How many different teams of 6 could be selected?

Working with Probability EquationsWhen in doubt, count it out!

1. Suppose you roll two dice simultaneously. What is the probability that number will the same on both dice?

Die 1 Die 2

1 1

2 2

3 3

4 4

5 5

6 6

𝑠𝑎𝑚𝑒 # 𝑟𝑜𝑙𝑙𝑒𝑑

𝑡𝑜𝑡𝑎𝑙 𝑟𝑜𝑙𝑙𝑠=

6

6 × 6=

6

36=1

6

Working with Probability EquationsWhen in doubt, count it out!

2. Suppose you roll two dice simultaneously. What is the probability that the sum will be greater than 9?

Die 1 Die 2 Sum

4 6 10

6 4 10

5 5 10

5 6 11

6 5 11

6 6 12

𝑠𝑢𝑚 > 9

𝑡𝑜𝑡𝑎𝑙 𝑟𝑜𝑙𝑙𝑠=

6

6 × 6=

6

36=1

6

Questions?

Working with Calculus EquationsWhat does each of these mean?

1. limx→5

x2 = 5 2 = 25

2.d

dxx3 = 3x3−1 = 3x2

3.𝑑

𝑑𝑥4x = 4x1−1 = 4x0 = 4 1 = 4

4.𝑑

𝑑𝑥x3 + x2 =

𝑑

𝑑𝑥x3 +

𝑑

𝑑𝑥x2 = 3x2 + 2x

5.𝑑

𝑑𝑥x3)(x2 =

𝑑

𝑑𝑥x3 x2 + x3

𝑑

𝑑𝑥x2

= 3x2 x2 + x3 2x = 3x4 + 2x4 = 5x4

Working with Calculus EquationsChain Rule! When to use?

Find the derivative of (6x2 + x)3.

Let u = 6x2 + x and y = u3.

𝑑𝑦

𝑑𝑢= 3u2

𝑑𝑢

𝑑𝑥= 12x + 1

𝑑𝑦

𝑑𝑥=𝑑𝑦

𝑑𝑢×𝑑𝑢

𝑑𝑥= 3u2 12x + 1 = 3 6x2 + x 2 12x + 1

= 3 36x4 + 12x3 + x2 12x + 1

= 3 432x5 + 144x4 + 12x3 + 36x4 + 12x3 + x2

= 1296x5 + 540x4 + 72x3 + 3x2

Working with Calculus EquationsWhat does each of these mean?

1. x−3 𝑑𝑥 =x−3+1

−3+1+ C = −

x−2

2+ C

2. +x−2) x2)𝑑𝑥 =x−2+1

−2+1+

x2+1

2+1+ C =

x−1

−1+

x3

3+ C = −x−1 +

x3

3+ C

3. 4x−3− 𝑑𝑥 = −4x−3+1

−3+1+ C = −4

x−2

−2+ C =

2

x2+ C

4. 3−32x2𝑑𝑥 =

2

3x3 + C ቤ

3−3

=2

33 3 −

2

3−3 3 = 18 − −18 = 36

Questions?

Addition & Subtraction

Multiplication & Division

Exponents & Logarithms

𝑥 + 2 = 10 𝑥 = 10 − 2

2𝑥 = 10 𝑥 =10

2

10𝑥 = 2 x = 𝑙o𝑔10 2

10𝐴 = 𝐵

𝑙𝑜𝑔 𝐵 = 𝐴

*PCAT logarithms are mostly base 10

𝑙𝑜𝑔10(10𝑥) = lo𝑔10(2)

Logarithms

If the pH of a solution is 4, what is the concentration of hydrogen ions in the solution?

− log 𝐻+ = 4 𝐻+ = 10−4

pX = –log [ X ]

What is the pKa of a compound if its Ka is 10−4?

− log[𝐾a] = pKa

log 𝐻+ = −4

𝑙𝑜𝑔 𝐵 = 𝐴 𝐵 = 10𝐴pKa, pH, pOH, pKb, etc…

− log 𝐻+ = 𝑝𝐻

− log[10−4] = 𝑝𝐾𝑎

𝑙𝑜𝑔 𝐵 = 𝐴 𝐵 = 10𝐴

log 10−4 = −pKa 10−4 = 10−pKa 𝑝𝐾𝑎 = 4

Logarithms

Log base e

𝑒𝐴 = 𝐵

ln 𝐵 = 𝐴

10𝐴 = 𝐵

𝑙𝑜𝑔 𝐵 = 𝐴

𝑒 ≈ 2.7

What is x in the equation 15 = –3 ln x?

ln 𝑥 =15

−315 = −3 ln 𝑥 ln 𝑥 = −5 𝑒−5 = 𝑥

𝒙 = 𝟎. 𝟎𝟎𝟔𝟕𝟒

ln 𝐵 = 𝐴 𝐵 = 𝑒𝐴

1

𝑒5= 𝑥

1

35< 𝑥 <

1

250.004 < 𝑥 < 0.031

Natural Log

5, 9, 9, 10, 13, 14Mean 5 + 9 + 9 + 10 + 13 + 14

6

60

6= 10

“Average”

Median 5, 9, 9, 10, 13, 14

“Middle”

Mode

5, 9, 9, 10, 13, 14“Most common”

9.5

9

For this set of numbers:

Mean = 10Median = 9.5Mode = 9

Numerical Analysis

5, 6, 8, 9, 9, 10, 11, 12, 12, 13, 14

Range

14 − 5 = 9

Standard Deviation

Both sets have a mean of 50, but set B has a much higher standard deviation.49, 50, 50, 51 4, 28, 72, 96

A B

Variance = SD2

Interquartile Range

2, 3, 7, 9, 9, 10, 14, 16, 19, 20, 21, 23

Q1 = 8 Q3 = 19.5

IQR = Q3 – Q1 = 19.5 – 8 = 11.5

Q2Q1 Q3

Numerical Analysis

P-value

Standard Deviation

R-squared (R2)

Likelihood that the null hypothesis could be true

How much the values are grouped around the mean

How closely correlated two items are.

Data Interpretation

Example: Students who take at least 3 practice exams score higher on the PCAT than those who don’t taken any practice exams (p = 0.05).

✓ There is a 5% probability that taking 3 practice exams does not affect PCAT scores.✓ There is a 95% probability that taking 3 practice exams does affect PCAT scores.

Small SD Large SD

R² = 0.98810

5

10

15

0 2 4 6

R² = 0.61220

5

10

15

0 2 4 6

Data Interpretation

What Next?Reflect on difficult topics of concern.

Office Hours: Open Q&A

• Identify equations or QR concepts that are difficult for you

• Find 10 practice problems using each equation or concept

• After practice, focus on mastering that topic improving your timing and efficiency

Think of a question after

Office Hours are over?

Post on our forums!

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