Question

determine whether the following individual events are independent or dependent. then find the probability of the combined event.

randomly drawing and immediately eating two red pieces of candy in a row from a bag that contains 10 red pieces of candy out of 40 pieces of candy total.

choose the correct answer below.
(round to three decimal places as needed.)

a. the individual events are dependent. the probability of the combined event is
b. the individual events are independent. the probability of the combined event is

Explanation:

Determine event dependence

Using the Independent Events knowledge point

Because the first piece of candy is eaten, it is not replaced before drawing the second piece. This action changes the total number of candies and the number of red candies remaining in the bag. Therefore, the outcome of the first draw affects the probability of the second draw, making the individual events dependent.

Calculate first draw probability

Using the Probability knowledge point

$$ P(\text{First Red}) = \frac{10}{40} = \frac{1}{4} = 0.25 $$

Calculate second draw probability

Since the first red candy was eaten, there are now \(10 - 1 = 9\) red candies left, and a total of \(40 - 1 = 39\) candies remaining in the bag.

$$ P(\text{Second Red} \mid \text{First Red}) = \frac{9}{39} = \frac{3}{13} $$

Calculate combined probability

The probability of drawing two red candies in a row is the product of the individual dependent probabilities:

$$ P(\text{Both Red}) = P(\text{First Red}) \times P(\text{Second Red} \mid \text{First Red}) = \frac{1}{4} \times \frac{3}{13} = \frac{3}{52} $$

$$ \frac{3}{52} \approx 0.05769 $$

Rounding to three decimal places yields \(0.058\).

Answer:

• (A) The individual events are dependent. The probability of the combined event is 0.058 (Correct answer)
• (B) The individual events are independent. The probability of the combined event is 0.063