# probability Exs

```At the Iron Bank, 36% of customers invest in Stocks, 22% invest in Futures, and 15% invest in both Stocks
and Futures. You could have used a contingency table to answer each part of this question. A contingency
table is perfectly sufficient to show work (and thus to earn partial credit on any incorrect responses).
P(Stocks) = 36%
P(Futures) = 22%
P(Stocks,Futures) = 15%
1- what proportion of Iron Bank customers invest in neither Stocks nor Futures? Use the
negation rule in combination with the addition rule.
P (neither stocks nor futures) = 1 −P (stocks or futures)
= 1 −[P (stocks + P (futures) −P (stocks, futures)]
= 1 −[0.36 + 0.22 −0.15]
= 1 −0.43
= 0.57
OR
2- if the Iron Bank customers who invest in Futures, what percentage invest in Stocks?
Recognize that this problem is asking for a conditional probability and use the multiplication
rule:
P (stocks |futures) = P (stocks, futures)/P (futures) =
0.15/0.22 ≈0.68
3- What is the probability that a randomly selected Iron Bank customer invests in Stocks and
does not invest in Futures?
P (stocks) = P (stocks, no futures) + P (stocks, futures)
0.36 = P (stocks, no futures) + 0.15
0.36 −0.15 = P (stocks, no futures) = 0.21
OR
Numerical V. Categorical
- Numerical variables are represented by a number with units.
Numerical variables have consistent intervals (dollars, years, percentage
points).
They answer the questions, “how much?” or “how many?”
-
Categorical variables are represented by a word, category or number without units.
you can’t do arithmetic with them (15 * 9 rank + 2 rank= 137 ranks?)
Categorical variables have no or inconsistent intervals (genre has no
intervals, rank has
inconsistent intervals).
Lesson 3 Counting
PROBABILITY NOTATION- The probability of some event is a number between
zero and one:
π ≤ π(π¬π¨π¦π ππ―ππ§π­) ≤ π
An event with probability 1 is certain to occur.
An event with probability 0 is certain not to occur.
For example:
- π (ππππ πππππ  βππππ ) = 0.5
- π (ππππβπ‘ ππππππ‘π  ππ π‘πππ) = 0.812
- π (ππππ πππ¦ ππ π»πππ) = 0.000000000000001
WHERE DO PROBABILITIES COME FROM?
- Careful counting
Ex- Draw a card at random from a standard 52-card deck
What is π (πππππππ)?
What is π (ππππ ππ ππ’πππ)?
π (πππππππ) = 13/52 = 0.25
π (ππππ ππ ππ’πππ) = 8/52 ≈ 0.154
THE COUNTING PRINCIPLE:
π = # ππ’π‘πππππ  ππ πππ‘ππππ π‘/ π‘ππ‘ππ # ππ ππ’π‘πππππ  = π/π
Key assumption: All outcomes are equally likely
Careful Counting Example 2:
Rolling two fair dice
Let π be the sum of the two numbers.
Clearly, π is random: it varies from one roll to the next.
What is π(π ≥ 9)?
π (π ≥ 9) = 10/36 ≈ 0.278
WHERE DO PROBABILITIES COME FROM?
-
Data
Ex- π(πππ€ππππ ππππ¦ ππ  π ππππ) = 100/206 ≈ 0.485
Ex- P (A college student will be in a car accident this year) ≈ 0.009
WHERE DO PROBABILITIES COME FROM?
-
Subjective judgment
Ex- P (ππππππ π€πππ  ππ ππππ) = ?
Ex- π (π΄ππππ π π‘πππ π’π πππ₯π‘ π¦πππ) = 0.7 ? (buy)
π (π΄ππππ π π‘πππ π’π πππ₯π‘ π¦πππ)= 0.4 ? (Sell)
WHERE DO PROBABILITIES COME FROM?
-
Other probabilities
- THREE RULES OF PROBABILITY
- Negation Rule
In general, for any event π΄: π π§π¨π­ π = π − π(π)
- Also known as the “complement rule”
Let’s pretend that the probability of rain tomorrow is 0.3
- Then clearly the probability that it will NOT rain tomorrow is 0.7
Say the probability that Serena Williams wins the U.S. Open tournament is 0.15.
- Thus, the probability that Serena does NOT win is 0.85.
- Consider two events A and B.
-Let π(π΄, π΅) be the joint probability that both A and B happen.
- Draw a card at random. What is (π ππππ ππ  π΄ππ ππ πππππ)?
Key insight: “Ace” and “Spade” are not mutually exclusive outcomes.
- Therefore, we cannot simply add π π΄ππ) + π(πππππ like we did with π (King or Queen)
because we will double count the Ace of Spades.
Here’s what we know... Of the 52 cards:
4 are aces: π (π΄ππ) = 4/52
13 are spades: π (πππππ) = 13/52
1 is both an ace and a spade: π (π΄ππ,πππππ)= 1/52
Total= π π΄ππ ππ πππππ = 4/52 + 13/52 − 1/52 = ππ/ππ ≈ 0.308
CONDITIONAL PROBABILITY
π· (π¨/π©) is the “probability of A given B”
-examples:
π( ππππ π‘βππ  πππ‘ππππππ /ππππ’ππ¦ π‘βππ  πππππππ)
π (ππ ππππ‘π  ππ / ππ πβπππ ππ¦ π π‘ππ’πβπππ€π ππ‘ βππππ‘πππ)
π (ππππππ‘ππ π‘π πππππππ π πβπππ/ πππππππ πΊππ΄ &gt; 3.6)
Amazon:P (buy organic dog food / bought GPS dog collar)
Netflix:P(watch πππ’ππ πΊππππ  / watch ππ‘ππππππ πβππππ )
Perhaps the single most important fact to remember about conditional
probabilities:
π(π΄|π΅)≠π(π΅|π΄)
WHERE DO (conditional probabilities) PROBABILITIES COME FROM?
EX- Mammograms
- P cancer = 15/200
- P die, cancer = 3/200
- P die|cancer = 3/15
In general, we can estimate π(π΄|π΅) as: