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Introduction to TV show Numb3rs. Here are the important people: Don (FBI) and Charlie (Prof.) Eppes Their dad Charlie’s colleagues Don’s agents. Determining a card’s color. If you guess a card’s color (black or red), you have a 50% probability of being right

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introduction to tv show numb3rs
Introduction to TV show Numb3rs

Here are the important people:

  • Don (FBI) and Charlie (Prof.) Eppes
  • Their dad
  • Charlie’s colleagues
  • Don’s agents
determining a card s color
Determining a card’s color

If you guess a card’s color (black or red), you have a 50% probability of being right

What if you guessed 25 times and got it wrong all the times?

Increasing Likelihood of Occurrence

0

.5

1

Probability:

The event

is very

unlikely

to occur.

The occurrence

of the event is

just as likely as

it is unlikely.

The event

is almost

certain

to occur.

child kidnappings
Child Kidnappings

56% of the children are found alive

In 90% the parents are responsible

It is far more likely to find the child alive with help from the FBI

  • Let’s assume twice as likely

We want to know

  • a) the probability the parents will ask for help
  • b) the probability the parents are responsible given they do not ask for help

Use conditional probability theory and Bayes’ Theorem

child kidnappings continued
Child Kidnappings continued

H= parents ask for help from the FBI

HC= parents do not ask for help from the FBI

We first want to know:

Let B denote the event of bringing the child back alive. Past history with these FBI cases indicate that with help the child was found alive in 95%:

Thus (using conditional probabilities):

P(B) = P(H)P(B|H) + P(HC)P(B|HC)

0.56 = P(H).95 + (1 – P(H)).475

P(H) = 0.18

a) P(H) = ?

P(B|HC) = .475

P(B|H) = .95

child kidnappings continued5
Child Kidnappings continued

b) P(R|HC) = ?

Now we want to know:

We apply Bayes’ theorem:

From past history we know that if one of the parents was responsible, they did not ask for help in 90% of the cases.

We revise the prior probabilities as follows:

P(H|R) = .10

P(HC|R) = .90

shooting chains
Shooting chains

Gang related shootings result in shooting chains of, on average, 2.8 shootings (i.e., the initial shooting, plus 1.8 additional shootings on average), with a standard deviation of 1.1.

Four shootings stand out, as they resulted in shooting chains of 4, 5, 6, and 7 shootings respectively.

Amita says: “Statistically that wouldn’t happen if he had chosen the victims at random”

Test this Hypothesis at the a =1% level.

shooting chains continued
Shooting chains continued

Determine the hypotheses.

a = .01

Specify the level of significance.

Compute the test statistic.

Compute the p –value.

P(z > 4.91) = .0000

a = .01, z.01 = 2.33

Or: Determine the critical value

p-value: .0000 < 0.01Critical: 4.91 > 2.33

Determine whether to reject H0.

slide8

Shooting chains continued

If the confidence interval contains the hypothesized value 0, do not reject H0. Otherwise, reject H0.

The 98% confidence interval for is

Because the hypothesized value for the population mean, 0 = 2.8, is not in this interval, the hypothesis-testing conclusion is that the null hypothesis (H0:  = 2.8), can be rejected.

shooting chains continued9
Shooting chains continued

What if we do not actually know the standard deviation of the population of shooting chains?

We would have to use the standard deviation of the sample (4, 5, 6, and 7, leads to s = 1.3)

We would also have to use a t distribution:

For a = .01 and d.f. = 3, t.01 = 4.54

For t = 4.18, using n – 1 = 3 d.f., the p–value equals 0.012

Now we cannot reject the null hypothesis of H0:  = 2.8 at the 1% level!

traffic accidents
Traffic accidents

The FBI found that 5 of 13 traffic accident victims were related to previous serious traffic accidents. If the population average is 40%, can we speak of a coincidence or not? Test with a = 5%.

Note: in population proportions, we need to make sure that both np and n(1 – p) are greater than 5:

13 x 0.4 = 5.2 > 5 & 13 x (1 – 0.4) = 7.8 > 5

Determine the hypotheses.

a = .05

Specify the level of significance.

traffic accidents continued
Traffic accidents continued

a common

error is using

pin this formula

Compute the test statistic.

Compute the p –value.

P(z> -0.11) > 0.50

a = .05 z.95 = 1.64

Or: Determine the critical value

p-value: .545 > 0.05Critical: -0.11 < 1.64

Determine whether to reject H0.

traffic accidents continued12
Traffic accidents continued

Some more investigation leads to a further 5 victims who are related to serious traffic accidents. What can we say now?

Compute the test statistic.

Compute the p –value.

P(z> 2.71) < 0.01

a = .05 z.95 = 1.64

Or: Determine the critical value

p-value: .0033 < 0.05Critical: 2.71 > 1.64

Determine whether to reject H0.

some other interesting video clips
Some other interesting video clips

Always go back to the Data

Game show

On dating…

Type I and Type II errors

Benford’s Law

Logistic Regression

some video clips on randomness
Some video clips on randomness

Randomness – random spacing is not equal

Randomness – raindrops on a sidewalk

  • Notice the conditional probability math on the glass behind Charlie!

Randomness – shuffling cards