4.3.1 Conditional probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred.

In other words, it is the probability of an event \( A \) happening, given that another event \( B \) has already occurred.

• It is denoted as \( P(A|B) \), which means the probability of event \( A \) occurring given that event \( B \) has already occurred.

• Mathematically, conditional probability is calculated using the following formula:

\( P(A|B)=\frac{P(A \ and \ B)}{P(B)} \)

• Where \( P (A \ and \ B) \) represents the probability of both events \( A \) and \( B \) occurring together

• \( P(B) \) represents the probability of event \( B \) occurring

• Conditional probability is an important concept in many fields, including statistics, probability theory, and machine learning, as it helps in understanding the relationships between events and making predictions based on the available information.

Worked example:

A box contains \( 20 \) packets of potato chips.

\( 6 \) packets contain barbecue flavored chips.

\( 10 \) packets contain salt flavored chips.

\( 4 \) packets contain chicken flavored chips.

Maria takes two packets at random without replacement.

Show that the probability that she takes two packets of salt flavored chips is \( \frac{9}{38} \).

Find the probability that the first packet is salt flavored.

\( P(1^{st} \ packet \ is \ salt \ flavored)=\frac{10}{20} \)

Find the probability that the second packet is salt flavored.

\( P(2^{nd} \ packet \ is \ salt \ flavored)=\frac{9}{19} \)

Find the probability that the first packet and the second packet is salt flavored.

\( \frac{10}{20} \times \frac{9}{19} = \frac{90}{380} \)

Simplify the answer

\( \frac{9}{38} \)