Question

an item is randomly drawn from a bag with 16 fruit cups, 3 vegetable cups, 14 fruit cans, and 7 vegetable cans.

event a: vegetable
event b: cup

\\(p(a \text{ or } b) = ? \\)

hint: \\(p(a \text{ or } b) = p(a) + p(b) - p(a \text{ and } b)\\)
round your answer to the nearest hundredth.

Explanation:

Calculate the total number of items

We first find the total number of items in the bag by summing all the categories.
Using the Theoretical Probability knowledge point:

$$ \text{Total items} = 16 \text{ (fruit cups)} + 3 \text{ (vegetable cups)} + 14 \text{ (fruit cans)} + 7 \text{ (vegetable cans)} $$

$$ \text{Total items} = 40 $$

Determine the probability of Event A

Event A is drawing a vegetable item. We find the total number of vegetable items and divide by the total number of items.
Using the Theoretical Probability knowledge point:

$$ \text{Vegetable items} = 3 \text{ (vegetable cups)} + 7 \text{ (vegetable cans)} = 10 $$

$$ P(A) = \frac{10}{40} = 0.25 $$

Determine the probability of Event B

Event B is drawing a cup. We find the total number of cups and divide by the total number of items.
Using the Theoretical Probability knowledge point:

$$ \text{Total cups} = 16 \text{ (fruit cups)} + 3 \text{ (vegetable cups)} = 19 $$

$$ P(B) = \frac{19}{40} = 0.475 $$

Determine the probability of Event A and B

The intersection represents drawing an item that is both a vegetable and a cup (vegetable cups).
Using the Overlapping Events knowledge point:

$$ \text{Vegetable cups} = 3 $$

$$ P(A \text{ and } B) = \frac{3}{40} = 0.075 $$

Apply the addition rule for overlapping events

We use the given formula to find the probability of drawing a vegetable or a cup.
Using the Overlapping Events knowledge point:

$$ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $$

$$ P(A \text{ or } B) = \frac{10}{40} + \frac{19}{40} - \frac{3}{40} = \frac{26}{40} = 0.65 $$

Rounding to the nearest hundredth gives \(0.65\).

Answer:

An item is randomly drawn from a bag with 16 fruit cups, 3 vegetable cups, 14 fruit cans, and 7 vegetable cans.
Event A: vegetable
Event B: cup
\(P(A \text{ or } B) =\) <blank>0.65</blank>