+157 Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, then turns 3 random lamps on. What is the probability that the leftmost lamp on the shelf is red, and the leftmost lamp which is turned on is also red?
Thanks a lot in advance!
+451 Analyzing the Problem
Breaking Down the Events:
Event 1: The leftmost lamp on the shelf is red.
Event 2: The leftmost lamp turned on is red.
Calculating Probabilities:
We'll use the concept of conditional probability to calculate the probability of both events happening.
Calculating the Probabilities
Probability of Event 1:
There are 3 red lamps out of a total of 6 lamps.
So, the probability of the leftmost lamp being red is 3/6 = 1/2.
Probability of Event 2 given Event 1:
If the leftmost lamp is red, there are now 2 red lamps and 3 blue lamps remaining.
The probability of the leftmost lamp turned on being red is 2/5.
Conditional Probability:
The probability of both events happening is the product of the probability of Event 1 and the conditional probability of Event 2 given Event 1.
Final Calculation:
Probability = (1/2) * (2/5) = 2/10 = 1/5
Therefore, the probability that the leftmost lamp on the shelf is red and the leftmost lamp turned on is also red is 1/5.
