Probability

What is probability?

Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain):

P(A) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}

For example, drawing a King from a standard 52-card deck: 4 of the 52 cards are Kings, so $P(\text{King}) = \dfrac{4}{52} = \dfrac{1}{13}$ .

Independent vs. dependent events

Two events are independentwhen one doesn't affect the other's outcome, like tossing a coin and rolling a die together:

P(\text{Head and 3}) = P(\text{Head}) \times P(3) = \dfrac{1}{2}\times\dfrac{1}{6} = \dfrac{1}{12}

Two events are dependent when one changes the odds of the other, like drawing two cards without replacement. Drawing a King, then a Queen from the remaining 51 cards:

P(\text{King then Queen}) = \dfrac{4}{52}\times\dfrac{4}{51} = \dfrac{4}{663}

Conditional probability

Conditional probability asks: given that event B already happened, what's the chance of event A?

P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}

Worked example

Out of 50 students surveyed, 35 play basketball, and of those, 25 are boys. What's the probability that a basketball player is a boy?

Solution:Let A = "plays basketball" and B = "is a boy". Then $P(A) = \dfrac{35}{50}$ and $P(A \text{ and } B) = \dfrac{25}{50}$ :

P(B \mid A) = \dfrac{25/50}{35/50} = \dfrac{25}{35} = \dfrac{5}{7} \approx 0.71

Probability

What is probability?

Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain):

P(A) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}

For example, drawing a King from a standard 52-card deck: 4 of the 52 cards are Kings, so $P(\text{King}) = \dfrac{4}{52} = \dfrac{1}{13}$ .

Independent vs. dependent events

Two events are independentwhen one doesn't affect the other's outcome, like tossing a coin and rolling a die together:

P(\text{Head and 3}) = P(\text{Head}) \times P(3) = \dfrac{1}{2}\times\dfrac{1}{6} = \dfrac{1}{12}

Two events are dependent when one changes the odds of the other, like drawing two cards without replacement. Drawing a King, then a Queen from the remaining 51 cards:

P(\text{King then Queen}) = \dfrac{4}{52}\times\dfrac{4}{51} = \dfrac{4}{663}

Conditional probability

Conditional probability asks: given that event B already happened, what's the chance of event A?

P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}

Worked example

Out of 50 students surveyed, 35 play basketball, and of those, 25 are boys. What's the probability that a basketball player is a boy?

Solution:Let A = "plays basketball" and B = "is a boy". Then $P(A) = \dfrac{35}{50}$ and $P(A \text{ and } B) = \dfrac{25}{50}$ :

P(B \mid A) = \dfrac{25/50}{35/50} = \dfrac{25}{35} = \dfrac{5}{7} \approx 0.71

Probability

What is probability?

Independent vs. dependent events

Conditional probability

Worked example

Continue learning

Probability

What is probability?

Independent vs. dependent events

Conditional probability

Worked example

Continue learning