Question 34·Easy·Probability and Conditional Probability
A bag contains 5 red marbles and 3 blue marbles. A marble is selected at random, its color is recorded, and then the marble is returned to the bag. A second marble is then selected at random.
What is the probability that both marbles selected are red?
For probability questions with repeated draws and replacement, first find the probability of the desired result on a single trial by using favorable outcomes / total outcomes. Replacement usually means each trial has the same probability and the trials are independent. When the question asks for two (or more) events to all happen (like "both red" or "three heads in a row"), multiply the single-trial probabilities together, then simplify the resulting fraction and check that it is between 0 and 1.
Hints
Count total and favorable outcomes for one draw
How many marbles are in the bag altogether, and how many of those are red? Use that to find the probability that one randomly chosen marble is red.
Consider what replacement does
Since the marble is put back after the first draw, does the total number of marbles or the number of red marbles change before the second draw? How does that affect the probability on the second draw?
Connect the two draws
You want the first draw to be red and the second draw to be red. For two independent events, what operation (add, subtract, multiply, or divide) do you use to combine their probabilities?
Write the probability expression
Once you know the probability of red on each draw, write an expression for the probability that both draws are red using the multiplication rule, and then simplify the fraction.
Desmos Guide
Verify the probability calculation
In Desmos, type (5/8)*(5/8) on a new line. Look at the simplified fraction that Desmos gives; that value is the probability that both marbles drawn are red with replacement.
Step-by-step Explanation
Find the probability of getting a red marble on one draw
There are 5 red marbles and 3 blue marbles, for a total of marbles.
So, on any single draw, the probability of drawing a red marble is
Understand the effect of replacement
After the first marble is drawn, it is returned to the bag. That means the bag again has 5 red and 3 blue marbles before the second draw.
So the probability of drawing a red marble on the second draw is also
Because the marble is replaced, the two draws are independent events: what happens on the first draw does not change the probabilities on the second draw.
Use the multiplication rule for independent events
For two independent events and , the probability that both happen is
Here:
- = "first marble is red" with probability ,
- = "second marble is red" with probability .
So the probability that both marbles are red is
So the correct answer is .