Probability

The basic formula for probability is:

$\mathrm{probability}=\frac{\mathrm{number\; of\; favorable\; outcomes}}{\mathrm{total\; number\; of\; possible\; outcomes}}$

For example, if you roll a fair six-sided die:

• The probability of rolling a 3 is $\frac{1}{6}$ (one favorable outcome out of six possible outcomes)
• The probability of rolling an even number is $\frac{3}{6}=\frac{1}{2}$ (three favorable outcomes: 2, 4, or 6)

Probabilities can be expressed as fractions, decimals, or percentages. To convert between them:

• Fraction to percentage: Multiply by 100%
• Percentage to fraction: Divide by 100%

For example:

• $\frac{1}{4}=0.25=25\%$
• $75\%=\frac{75}{100}=\frac{3}{4}$

The sum of all possible probabilities for a given situation must equal 1 (or 100%). This is because one of the possible outcomes must occur.

For example, when rolling a die:

• Probability of rolling a 1: $\frac{1}{6}$
• Probability of rolling a 2: $\frac{1}{6}$
• Probability of rolling a 3: $\frac{1}{6}$
• Probability of rolling a 4: $\frac{1}{6}$
• Probability of rolling a 5: $\frac{1}{6}$
• Probability of rolling a 6: $\frac{1}{6}$
• Total probability: $\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=1$

A two-way table (also called a contingency table) shows the frequency of outcomes for two categorical variables. The rows represent one variable, and the columns represent another variable.

For example, consider a survey of 100 students about their favorite subject and grade level:

From this table, you can calculate various probabilities:

• Probability that a randomly selected student likes Math: $\frac{40}{100}=0.4=40\%$
• Probability that a randomly selected student is in 9th grade: $\frac{30}{100}=0.3=30\%$
• Probability that a randomly selected student is in 9th grade AND likes Math: $\frac{15}{100}=0.15=15\%$

The formula for conditional probability is:

$P\left(A\right|B)=\frac{P(AandB)}{P\left(B\right)}$

Using our previous example of the student survey:

• What is the probability that a student likes Math given that they are in 9th grade?
• P(Math|9th Grade) = $\frac{P(\mathrm{Math}and\mathrm{9th\; Grade})}{P\left(\mathrm{9th\; Grade}\right)}=\frac{\frac{15}{100}}{\frac{30}{100}}=\frac{15}{30}=\frac{1}{2}=0.5=50\%$

This means that 50% of 9th graders like Math.

Two events are independent if the occurrence of one event does not affect the probability of the other event. For independent events A and B:

$P\left(A\right|B)=P(A)$

For example, if you roll a die twice, the outcome of the first roll does not affect the outcome of the second roll. These are independent events.

Two events are dependent if the occurrence of one event affects the probability of the other event. For example, drawing cards from a deck without replacement: the probability of drawing a second ace depends on whether an ace was drawn first.

Example 1: A bag contains 3 red marbles, 4 blue marbles, and 5 green marbles. If a marble is randomly selected, what is the probability that it is blue?

Solution:

• Total number of marbles = 3 + 4 + 5 = 12
• Number of blue marbles = 4
• Probability = $\frac{4}{12}=\frac{1}{3}\approx 33.3\%$

Example 2: A standard deck of 52 cards contains 4 aces. If two cards are drawn without replacement, what is the probability that both cards are aces?

Solution:

• Probability of drawing first ace = $\frac{4}{52}=\frac{1}{13}$
• After drawing first ace, 3 aces remain in 51 cards
• Probability of drawing second ace = $\frac{3}{51}=\frac{1}{17}$
• Probability of both events = $\frac{1}{13}\times \frac{1}{17}=\frac{1}{221}\approx 0.0045\approx 0.45\%$