Explore: Conditional Probability Solutions

Conditional probability, $P (B | A)$ , is the probability of B, after A has occurred .
 The formula for conditional probability of dependent events is:
$P (B | A) = \frac{P (A \cap B)}{P (A)}$  
Because the order in which events occur matters, $P (B | A) \neq P (A | B) .$  
Conditional probability is useful when the conditions being compared are changed .
This allows you to compare events to a specific group rather than only to the total number of events.  
The words if, when, after, and given that often indicate there is a specific condition that must be considered.

Example Conditional Probability Phrases:

 When solving problems with conditional events: 
- Substitute what you know into the formula: 
  $P (A and B) = P (A) \cdot P (B | A)$ or $P (A and B) = P (B) \cdot P (A | B)$
-  Then solve for the missing term algebraically.

Example 3

A fruit bowl contains ten apples and eight nectarines. You decide to grab two pieces of fruit without looking. Determine the probability as a simplified fraction that:

Both pieces of fruit are apples.

$P (A | A) = \frac{10}{18} \cdot \frac{9}{17} = \frac{5}{9} \cdot \frac{9}{17} = \frac{5}{17}$

You grab a nectarine after you already have an apple.

$P (N | A) = \frac{8}{17}$

Note

Since you already know that you have an apple, you only need to find the probability of selecting a nectarine with one less piece of fruit.

Example 4

The employees of Buzz Coffee tracked orders on a busy morning. One of the employees created a Venn diagram to organize the orders and calculated some of the probabilities.

$P (Latte) = 0.4967 P (Muffin) = 0.4 P (Coffee) = 0.1333 P (Latte and Muffin) = 0.2167 P (Coffee and Muffin) = 0.0267 P (None) = 0.1867$

Determine if the orders are independent events.
Independent: $P (A and B) = P (A) \cdot P (B)$

Note

$P (Latte) = \frac{149}{300} P (Muffin) = \frac{120}{300} P (Coffee) = \frac{40}{300} P (Latte and Muffin) = \frac{65}{300} P (Coffee and Muffin) = \frac{8}{300} P (None) = 0.1867 = \frac{56}{300}$

Latte and Muffin

$P (L and M) ? P (L) \cdot P (M) 0.2167 \neq 0.4967 \cdot 0.4$

Coffee and Muffin

$P (C and M) ? P (C) \cdot P (M) 0.0267 \neq 0.1333 \cdot 0.4$

The events are not independent because $P (A and B) \neq P (A) \cdot P (B) .$

Find the probability that:

A customer ordered a muffin after they ordered a latte.

P(M | L) $= \frac{P (M and L)}{P (L)} = \frac{0.2167}{0.4967} = 0.4363$

A customer ordered a latte when they ordered a muffin.

P(L | M) $= \frac{P (L and M)}{P (M)} = \frac{0.2167}{0.4} = 0.54175$

Why are the results of the probabilities in parts B and C different?

Sample: The results of parts B and C are different because the order in which the events occur matters. (From the notes: $P (B | A) \neq P (A | B)$ )

Example 5

On a given day, the probability that Adrian will wear sneakers is 60%. The probability that Adrian wears a baseball hat is 40%. The likelihood that she wears sneakers if she is already wearing a hat is 30%.

Determine the probability of Adrian wearing a hat and sneakers.

$\begin{array}{rcl} P (S and H) & = & P (H) \cdot P (S | H) \\ = & (0.4) (0.3) \\ = & 0.12 \end{array}$

12% chance

Determine the probability of Adrian wearing a hat if she is already wearing sneakers.

$\begin{array}{rcl} P (S and H) & = & P (S) \cdot P (H | S) \\ \frac{0.12}{0.6} & = & \frac{0.6 \cdot P (H | S)}{0.6} \\ 0.2 & = & P (H | S) \end{array}$

20% chance