Probability - Examples - W04-06

Bernoulli process

24 - Binomial variable counting ones in repeated die rolls

A standard die is rolled 6 times. Use a binomial variable to find the probability of rolling at least 4 ones.

Solution

• & Labels
  • Let .
  • Interpret: counts the ones appearing over 6 rolls.
  • We seek .
• && Calculation
  • Exclusive events:

25 - Roll die until

Roll a fair die repeatedly. Find the probabilities that:

• (a) At most 2 threes occur in the first 5 rolls.
• (b) There is no three in the first 4 rolls, using a geometric variable.

Solution (a)

• & Labels.
  • Use to count the number of threes among the first six rolls.
  • Seek as the answer.
• && Calculations.
  • Divide into exclusive events:

(b)

• & Labels.
  • Use to give the roll number of the first time a three is rolled.
  • Seek as the answer.
• && Sum the PMF formula for .
  • Compute:
• !! Geometric series formula.
  • For any geometric series:
  • Apply formula:
• & Final answer is .

26 - Cubs winning the World Series

Suppose the Cubs are playing the Yankees for the World Series. The first team to 4 wins in 7 games wins the series. What is the probability that the Cubs win the series?

Assume that for any given game the probability of the Cubs winning is and losing is .

Solution (a) Using a binomial distribution

• & Label.
  • Let .
  • Thus is the probability that the Cubs win exactly 4 games over 7 played.
  • Seek as the answer.
• &&& Calculate.
  • Use binomial PMF:
  • Insert data:
  • Compute:
  • Convert :

(b) Using a Pascal distribution

• & Label.

  • Let .
  • Thus is the probability that the Cubs win their game on game number .
  • Seek as the answer.

• &&& Calculate.

  • Use Pascal PMF:
  • Insert data:
  • Compute:
  • Convert :

• !!! The algebra seems very different, right up to the end!

Expectation and variance

27 - Gambling game - tokens in bins

Consider a game like this: a coin is flipped; if then draw a token from Bin 1, if then from Bin 2.

• Bin 1 contents: 1 token $1,000, and 9 tokens $1
• Bin 2 contents: 5 tokens $50, and 5 tokens $1

It costs $50 to enter the game. Should you play it? (A lot of times?) How much would you pay to play?

Solution

28 - Expected value - rolling dice

Let be a random variable counting the number of dots given by rolling a single die.

Then: Let be an RV that counts the dots on a roll of two dice.

The PMF of : Then:

• ! Notice that .
  • In general, .
  • Let be a green die and a red die.
  • From the earlier calculation, and .
  • Since , we derive by simple addition!

29 - Expectation from PMF of related

Let have distribution given by this PMF:

Find .

Solution

• &&& Compute the PMF of .
  • PMF arranged by possible value:
• & Calculate the expectation.
  • Using formula for discrete PMF:

30 - Variance for composite using PMF and simpler formula

Suppose has this PMF:

	1	2	3
Find using the formula with .

(Hint: you should find and along the way.)

Poisson process

31 - Poisson calculation

Suppose . Find . (Leave the answer in exact form.)

Solution

• Conditioning definition:
• Expand numerator:
• Simplify:
• Compute for denominator:

32 - Arrivals at a post office

Client arrivals at a post office are modelled well using a Poisson variable.

Each potential client has a very low and independent chance of coming to the post office, but there are many thousands of potential clients, so the arrivals at the office actually come in moderate number.

Suppose the average rate is 5 clients per hour.

• (a) Find the probability that nobody comes in the first 10 minutes of opening. (The cashier is considering being late by 10 minutes to run an errand on the way to work.)
• (b) Find the probability that 5 clients come in the first hour. (I.e. the average is achieved.)
• (c) Find the probability that 9 clients come in the first two hours.

Solution (a)

• & Convert rate for desired window.
  • Expect clients every 10 minutes.
  • Let .
  • Seek as the answer.
• & Compute.
  • Formula:
  • Insert data and compute:

(b)

• & Rate is already correct.
  • Let .
  • Compute the answer:

(c)

• & Convert rate for desired window

  • Expect 10 clients every 2 hours.
  • Let .
  • Compute the answer:

• !!! Notice that 0.125 is smaller than 0.175.

Function on a random variable

33 - Expectation of function on RV given by chart

Suppose that in such a way that and and and no other values are mapped to .

	1	2	3
	4	1	87
Then:
And:
Therefore:

34 - Variance of uniform random variable

The uniform random variable on has distribution given by when .

• (a) Find using the shorter formula.
• (b) Find using “squaring the scale factor.”
• (c) Find directly.

Solution (a)

1. & Compute density.
   • The density for is:
2. & Compute and directly using integral formulas.
   • Compute :
   • Now compute :
3. & Find variance using short formula.
   • Plug in:

(b)

• “Squaring the scale factor” formula:
• Plugging in:

(c)

1. & Density.
   • The variable will have the density spread over the interval .
   • Density is then:
2. & Plug into prior variance formula.
   • Use and .
   • Get variance:
   • Simplify:

35 - PDF of derived from CDF

Suppose that .

• (a) Find the PDF of .
• (b) Find the PDF of .

Solution (a)

• Formula:
• Plug in:

(b)

• By definition:
• Since is increasing, we know:
• Therefore:
• Then using differentiation:

36 - Probabilities via CDF

Suppose the CDF of is given by . Compute:

• (a)
• (b)
• (c)
• (d)

Solution

Continuous wait times

37 - Earthquake wait time

Suppose the San Andreas fault produces major earthquakes modeled by a Poisson process, with an average of 1 major earthquake every 100 years.

• (a) What is the probability that there will not be a major earthquake in the next 20 years?
• (b) What is the probability that three earthquakes will strike within the next 20 years?

Solution (a) Since the average wait time is 100 years, we set earthquakes per year. Set and compute:

(b) The same Poisson process has the same earthquakes per year. Set , so: and compute: