# 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:
𝟎 ≤ 𝐏(𝐬𝐨𝐦𝐞 𝐞𝐯𝐞𝐧𝐭) ≤ 𝟏
An event with probability 1 is certain to occur.
An event with probability 0 is certain not to occur.
For example:
- 𝑃 (𝑐𝑜𝑖𝑛 𝑙𝑎𝑛𝑑𝑠 ℎ𝑒𝑎𝑑𝑠) = 0.5
- 𝑃 (𝑓𝑙𝑖𝑔ℎ𝑡 𝑑𝑒𝑝𝑎𝑟𝑡𝑠 𝑜𝑛 𝑡𝑖𝑚𝑒) = 0.812
- 𝑃 (𝑐𝑜𝑙𝑑 𝑑𝑎𝑦 𝑖𝑛 𝐻𝑒𝑙𝑙) = 0.000000000000001
WHERE DO PROBABILITIES COME FROM?
- Careful counting
Ex- Draw a card at random from a standard 52-card deck
What is 𝑃 (𝑑𝑖𝑎𝑚𝑜𝑛𝑑)?
What is 𝑃 (𝑘𝑖𝑛𝑔 𝑜𝑟 𝑞𝑢𝑒𝑒𝑛)?
𝑃 (𝑑𝑖𝑎𝑚𝑜𝑛𝑑) = 13/52 = 0.25
𝑃 (𝑘𝑖𝑛𝑔 𝑜𝑟 𝑞𝑢𝑒𝑒𝑛) = 8/52 ≈ 0.154
THE COUNTING PRINCIPLE:
𝑃 = # 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡/ 𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 𝑌/𝑁
Key assumption: All outcomes are equally likely
Careful Counting Example 2:
Rolling two fair dice
Let 𝑆 be the sum of the two numbers.
Clearly, 𝑆 is random: it varies from one roll to the next.
What is 𝑃(𝑆 ≥ 9)?
𝑃 (𝑆 ≥ 9) = 10/36 ≈ 0.278
WHERE DO PROBABILITIES COME FROM?
-
Data
Ex- 𝑃(𝑛𝑒𝑤𝑏𝑜𝑟𝑛 𝑏𝑎𝑏𝑦 𝑖𝑠 𝑎 𝑔𝑖𝑟𝑙) = 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 (𝑆𝑒𝑟𝑒𝑛𝑎 𝑤𝑖𝑛𝑠 𝑈𝑆 𝑂𝑝𝑒𝑛) = ?
Ex- 𝑃 (𝐴𝑝𝑝𝑙𝑒 𝑠𝑡𝑜𝑐𝑘 𝑢𝑝 𝑛𝑒𝑥𝑡 𝑦𝑒𝑎𝑟) = 0.7 ? (buy)
𝑃 (𝐴𝑝𝑝𝑙𝑒 𝑠𝑡𝑜𝑐𝑘 𝑢𝑝 𝑛𝑒𝑥𝑡 𝑦𝑒𝑎𝑟)= 0.4 ? (Sell)
WHERE DO PROBABILITIES COME FROM?
-
Other probabilities
- THREE RULES OF PROBABILITY
- Negation Rule
In general, for any event 𝐴: 𝐏 𝐧𝐨𝐭 𝐀 = 𝟏 − 𝐏(𝐀)
- 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 𝑃(𝐴, 𝐵) be the joint probability that both A and B happen.
Then 𝑷(𝑨𝒐𝒓𝑩) = 𝑷(𝑨) +𝑷 (𝑩) −𝑷(𝑨,𝑩)
- Draw a card at random. What is (𝑃 𝑐𝑎𝑟𝑑 𝑖𝑠 𝐴𝑐𝑒 𝑜𝑟 𝑆𝑝𝑎𝑑𝑒)?
Key insight: “Ace” and “Spade” are not mutually exclusive outcomes.
- Therefore, we cannot simply add 𝑃 𝐴𝑐𝑒) + 𝑃(𝑆𝑝𝑎𝑑𝑒 like we did with 𝑃 (King or Queen)
because we will double count the Ace of Spades.
Here’s what we know... Of the 52 cards:
4 are aces: 𝑃 (𝐴𝑐𝑒) = 4/52
13 are spades: 𝑃 (𝑆𝑝𝑎𝑑𝑒) = 13/52
1 is both an ace and a spade: 𝑃 (𝐴𝑐𝑒,𝑆𝑝𝑎𝑑𝑒)= 1/52
Total= 𝑃 𝐴𝑐𝑒 𝑜𝑟 𝑆𝑝𝑎𝑑𝑒 = 4/52 + 13/52 − 1/52 = 𝟏𝟔/𝟓𝟐 ≈ 0.308
CONDITIONAL PROBABILITY
𝑷 (𝑨/𝑩) is the “probability of A given B”
-examples:
𝑃( 𝑟𝑎𝑖𝑛 𝑡ℎ𝑖𝑠 𝑎𝑓𝑡𝑒𝑟𝑛𝑜𝑜𝑛 /𝑐𝑙𝑜𝑢𝑑𝑦 𝑡ℎ𝑖𝑠 𝑚𝑜𝑟𝑛𝑖𝑛𝑔)
𝑃 (𝑈𝑇 𝑏𝑒𝑎𝑡𝑠 𝑂𝑈 / 𝑈𝑇 𝑎ℎ𝑒𝑎𝑑 𝑏𝑦 𝑎 𝑡𝑜𝑢𝑐ℎ𝑑𝑜𝑤𝑛 𝑎𝑡 ℎ𝑎𝑙𝑓𝑡𝑖𝑚𝑒)
𝑃 (𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑡𝑜 𝑚𝑒𝑑𝑖𝑐𝑎𝑙 𝑠𝑐ℎ𝑜𝑜𝑙/ 𝑐𝑜𝑙𝑙𝑒𝑔𝑒 𝐺𝑃𝐴 &gt; 3.6)
Amazon:P (buy organic dog food / bought GPS dog collar)
Netflix:P(watch 𝑆𝑞𝑢𝑖𝑑 𝐺𝑎𝑚𝑒𝑠 / watch 𝑆𝑡𝑟𝑎𝑛𝑔𝑒𝑟 𝑇ℎ𝑖𝑛𝑔𝑠)
Perhaps the single most important fact to remember about conditional
probabilities:
𝑃(𝐴|𝐵)≠𝑃(𝐵|𝐴)
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 𝑃(𝐴|𝐵) as:
𝑷 (𝑨/𝑩) = (𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝑨 𝒂𝒏𝒅 𝑩 𝒃𝒐𝒕𝒉 𝒉𝒂𝒑𝒑𝒆𝒏𝒊𝒏𝒈 / 𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝑩 𝒉𝒂𝒑𝒑𝒆𝒏𝒊𝒏𝒈)
MULTIPLICATION RULE
The probability of A given B is equal to the frequency of A and B both
happening divided by the frequency of B happen.
- The multiplication rule expresses this idea in general terms:
𝑷 (𝑨/𝑩) = 𝑷(𝑨,𝑩)/ 𝑷(𝑩)
- We can also use the alternate version below if we want to go in
reverse, from a conditional probability to a joint probability.
- This says the same thing with the terms re-arranged:
𝑷(𝑨,𝑩) =𝑷(𝑨/𝑩∗𝑷(𝑩)
-
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