Probability
1. Introduction to Probability:
Probability is the measure of how likely an event is to occur. It is always a number between
0 and 1, where:
- 0 means an event will not happen.
- 1 means an event is certain to happen.
- Any value between 0 and 1 represents the likelihood of the event occurring.
Formula: P(A) = (Number of favorable outcomes) / (Total number of possible outcomes)
Example: If you roll a fair six-sided die, the probability of getting a 4 is:
P(4) = 1/6 because there is one favorable outcome (getting a 4) and six possible outcomes
(1, 2, 3, 4, 5, 6).
2. Complementary Events:
Complementary events are two events where one event happening means the other cannot
happen. The probability of an event and its complement always add up to 1.
Example: If the probability of rolling a 3 is P(3) = 1/6, then the probability of not rolling a 3
(the complement) is:
P(not 3) = 1 - 1/6 = 5/6.
3. Equally Likely Outcomes:
When all outcomes of an event are equally likely, the probability is calculated using the
formula:
P(A) = (Number of favorable outcomes) / (Total number of equally likely outcomes)
Example 1: Tossing a Coin:
There are 2 possible outcomes: heads (H) or tails (T).
The probability of getting a tail is P(tail) = 1/2.
Example 2: Rolling a Fair Die (Even Numbers):
The even numbers are 2, 4, and 6, so there are 3 favorable outcomes.
The probability of rolling an even number is P(even) = 3/6 = 1/2.
4. Using Lists to Show Equally Likely Outcomes:
Example 1: Rolling Two Dice:
What is the probability of rolling a sum of 7 when rolling two fair dice?
List of all possible outcomes:
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
...
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
The outcomes where the sum is 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
The probability of rolling a sum of 7 is P(sum=7) = 6/36 = 1/6.
5. Tree Diagrams:
Example 1: Tossing Two Coins
The possible outcomes for two coin tosses can be represented by a tree diagram:
Coin 1
/ \
H T
/ \ / \
H T H T
The possible outcomes are: HH, HT, TH, TT.
The probability of getting exactly one head (HT or TH) is P(one head) = 2/4 = 1/2.
6. Experimental vs Theoretical Probability:
Example 1: Experimental Probability of Rolling a 4
- Theoretical Probability: The probability of rolling a 4 on a fair six-sided die is P(4) = 1/6.
- Experimental Probability: Suppose you roll the die 30 times and get the following results:
5 times a 4.
The experimental probability is P(4) = 5/30 = 1/6.
Notice that the experimental probability matches the theoretical probability.
7. Comparing Probabilities (Experimental vs Theoretical):
Example 1: Experimental vs Theoretical Probability of Drawing a Red Card from a Deck
- Theoretical Probability: A standard deck has 52 cards, 26 of which are red (13 hearts and
13 diamonds). The probability of drawing a red card is P(red card) = 26/52 = 1/2.
- Experimental Probability: Suppose you draw 100 cards with replacement and get 52 red
cards. The experimental probability is P(red card) = 52/100 = 0.52.
As the number of trials increases, the experimental probability will approach the theoretical
probability.
Difference Between Theoretical and Experimental Probability
Feature Theoretical Probability Experimental Probability
Based on Mathematical calculations Real-life trials
Formula P(A) = Favorable outcomes / P(A) = Times event occurs /
Total outcomes Total trials
Accuracy Assumes perfect randomness Can vary due to chance
Examples:
1. Drawing an ace from a deck of 52 cards:
- Theoretical: P(Ace) = 4/52 = 1/13
- Experimental: If 52 cards are drawn and 5 aces appear, P(Ace) = 5/52
2. Flipping a coin:
- Theoretical: Probability of heads = 0.5
- Experimental: If flipped 10 times and heads appears 7 times, P(Heads) = 0.7 due to
randomness
