The document discusses probability and provides examples to explain key concepts such as sample space, probability calculations, independent and dependent events, odds, and more. Probability is defined as the chance of an event occurring and is calculated by taking the number of outcomes in the event and dividing by the total number of possible outcomes. A variety of examples using coins, cards, and dice help illustrate how to determine probabilities and odds for different scenarios.

1. PROBABLY PROBABILITY
Tackle the Math
Introduction to Probability

2. Probability
■ Probability is a part of math dealing with the chance of an event occurring.
■ Value between 0 and 1.
■ A probability of 0 means that an event is impossible
■ A probability of 1 means that an event is guaranteed
■ The higher the probability, the more likely the event is to occur.

3. Probability Definitions
■ An event is a set of outcomes that are possible to occur from an experiment (A).
■ Sample space is the set of all possible outcomes (S).
■ Calculating probability:
– The probability of an event “P(A)”
– 𝑃 𝐴 =
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴
𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 (𝑆)
■ Let’s look at some examples!
S
A

4. Example 1
You have a standard fair coin (equal chance of getting a head or tail).
(a) Write the sample space for flipping the coin one time.
(b) Find the probability of flipping a head in one toss.
𝑆 = {𝐻, 𝑇}
𝐴 = {𝐻}
𝑃 𝐴 = 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
𝑇𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 (𝑆)
=
1
2

5. Example 1 (continued)
You have a standard fair coin (equal chance of getting a head or tail).
(c)Write the sample space for flipping the coin 2 times.
(d) Find the probability of flipping one head and one tail in 2 flips.
𝑆 = {𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇}
𝐴 = {𝐻𝑇, 𝑇𝐻}
𝑃 𝐴 = 𝑃 1𝐻 𝑎𝑛𝑑 1𝑇 =
2
4
=
1
2

6. Independence
■ 2 events are independent if the occurrence of one does not affect the
probability of the other.
– E.g. The outcome of flipping one coin does not affect the outcome of
flipping another coin.
– For events A and B, 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) × 𝑃(𝐵)
■ 2 events are dependent if the occurrence of one does affect the probability of
the other.
– E.g. If you draw 2 cards from a deck without replacing.
– For events A and B, 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) × 𝑃(𝐵 𝑎𝑓𝑡𝑒𝑟 𝐴)

7. Example 2
You have a standard deck of 52 cards
(a) What is the probability of pulling an ace?
(b) Replacing the first card pulled, what is the probability of pulling a club?
𝐴 = {𝑎𝑐𝑒 𝑜𝑓 𝑠𝑝𝑎𝑑𝑒𝑠, 𝑎𝑐𝑒 𝑜𝑓 ℎ𝑒𝑎𝑟𝑡𝑠, 𝑎𝑐𝑒 𝑜𝑓 𝑐𝑙𝑢𝑏𝑠, 𝑎𝑐𝑒 𝑜𝑓 𝑑𝑖𝑎𝑚𝑜𝑛𝑑𝑠}
𝑃 𝐴 = 𝑃 𝑎𝑐𝑒 =
# 𝑜𝑓 𝑎𝑐𝑒𝑠
# 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 𝑖𝑛 𝑑𝑒𝑐𝑘
=
4
52
=
1
13
𝐹𝑖𝑟𝑠𝑡, 𝑟𝑒𝑐𝑜𝑔𝑛𝑖𝑧𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒𝑠𝑒 𝑒𝑣𝑒𝑛𝑡𝑠 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡.
𝐴 = {2𝐶, 3𝐶, 4𝐶, 5𝐶, 6𝐶, 7𝐶, 8𝐶, 9𝐶, 10𝐶, 𝐽𝐶, 𝑄𝐶, 𝐾𝐶, 𝐴𝐶}
𝑃 𝐴 = 𝑃 𝑐𝑙𝑢𝑏 =
# 𝑜𝑓 𝑐𝑙𝑢𝑏𝑠 𝑖𝑛 𝑑𝑒𝑐𝑘
# 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 𝑖𝑛 𝑑𝑒𝑐𝑘
=
13
52
=
1
4

8. Example 2 (continued)
You have a standard deck of 52 cards
(c)What is the probability of first pulling a 2 and then pulling a face card
without replacing the first card?
𝐴 = {2𝐻, 2𝐶, 2𝑆, 2𝐷}
𝐵 = {𝐽𝐻, 𝐽𝐶, 𝐽𝑆, 𝐽𝐷, 𝑄𝐻, 𝑄𝐶, 𝑄𝑆, 𝑄𝐷, 𝐾𝐻, 𝐾𝐶, 𝐾𝑆, 𝐾𝐷}
𝐸𝑣𝑒𝑛𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 × 𝑃 𝐵 𝑎𝑓𝑡𝑒𝑟 𝐴 = 𝑃 𝑡𝑤𝑜 × 𝑃 𝑓𝑎𝑐𝑒 𝑎𝑓𝑡𝑒𝑟 𝑝𝑢𝑙𝑙𝑖𝑛𝑔 1 𝑡𝑤𝑜
=
4
52
×
12
51
=
1
13
×
4
17
=
4
221
≈ 1.81%

9. Example 2 (continued)
You have a standard deck of 52 cards
(d)What is the probability of pulling 2 face card in a row without replacing
the first card?
𝐴 = {𝐽𝐻, 𝐽𝐶, 𝐽𝑆, 𝐽𝐷, 𝑄𝐻, 𝑄𝐶, 𝑄𝑆, 𝑄𝐷, 𝐾𝐻, 𝐾𝐶, 𝐾𝑆, 𝐾𝐷}
𝐵 = {11 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠}
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 × 𝑃 𝐵 𝑎𝑓𝑡𝑒𝑟 𝐴 = 𝑃 𝑓𝑎𝑐𝑒 × 𝑃 𝑓𝑎𝑐𝑒 𝑎𝑓𝑡𝑒𝑟 𝑝𝑢𝑙𝑙𝑖𝑛𝑔 1 𝑓𝑎𝑐𝑒
=
12
52
×
11
51
=
3
13
×
11
51
=
11
221
≈ 4.98%

10. Odds
■ Odds provide a measure of likelihood of an event (A).
■ Recall: 𝑃 𝐴 =
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴
𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 (𝑆)
, where A is an event and S is the sample
space.
■ 𝑂𝑑𝑑𝑠 𝐴 =
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑛𝑜𝑡 𝑖𝑛 𝐴
=
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴
𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑆 − # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴
■ Also written as = 𝐴: 𝑛𝑜𝑡 𝐴
■ Odds = 1: An event is equally likely to occur or not occur
■ Odds > 1: An event is more likely to occur than not occur
■ Odds < 1: An event is less likely to occur

11. Example 3
You have a fair 6-sided die.
(a) What are the odds of rolling a 5?
(b)What are the odds of rolling an even number?
𝐴 = {5}
𝑂𝑑𝑑𝑠 𝐴 = 𝑂𝑑𝑑𝑠 𝑟𝑜𝑙𝑙𝑖𝑛𝑔 5 =
# 𝑜𝑓5𝑠
# 𝑜𝑓 𝑛𝑜𝑡 5𝑠
=
1
5
or 1: 5
𝐴 = {2,4,6}
𝑂𝑑𝑑𝑠 𝐴 = 𝑂𝑑𝑑𝑠 𝑟𝑜𝑙𝑙𝑖𝑛𝑔 𝑒𝑣𝑒𝑛 =
# 𝑜𝑓𝑒𝑣𝑒𝑛𝑠
# 𝑜𝑓 𝑜𝑑𝑑𝑠
=
3
3
= 1 or 1: 1

12. Your Turn
You have 2 fair, 6-sided dice.
(a) What is the probability that both dice show even numbers?
(b)What is the probability that the 2 dice add up to 8?

13. UPCOMING
EVENTS
■ Tackle the Math Series
– Why Does Order Matter?
– 4 Out of 3 People Struggle
with Math
– Probably Probability
– Do You Know the Line?
– Beating the System (of
Equations)

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