Answer:
A. The probability that both beads are red is 5/92.
B. The probability that one bead is black and the other is yellow is 6/23.
Step-by-step explanation: To find the probabilities, we need to calculate the total number of possible outcomes and the number of favorable outcomes for each scenario.
A. Both beads are red:
Total number of beads = 6 + 8 + 10 = 24
The first bead can be any of the 24 beads. After picking the first bead, there will be 5 red beads left out of the remaining 23 beads. Thus, the probability of picking a red bead is:
P(Red on first pick) = 6/24 = 1/4
For the second pick, there will be one less bead, and one less red bead. Therefore, the probability of picking another red bead is:
P(Red on second pick) = 5/23
To find the probability of both events occurring, we multiply the probabilities together:
P(Both red) = P(Red on first pick) * P(Red on second pick)
= (1/4) * (5/23)
= 5/92
Therefore, the probability that both beads are red is 5/92.
B. One bead is black and the other is yellow:
Total number of beads = 24 (same as before)
The first bead can be any of the 24 beads. After picking the first bead, there will be 8 black beads and 10 yellow beads remaining. Thus, the probability of picking a black bead is:
P(Black on first pick) = 8/24 = 1/3
For the second pick, there will be one less bead, and if the first bead was black, there will be 10 yellow beads left. If the first bead was yellow, there will be 8 black beads left. Therefore, the probability of picking a yellow bead after a black bead is:
P(Yellow on second pick after black) = 10/23
However, we also need to consider the case when the first bead is yellow and the second bead is black. In that case, the probability is:
P(Black on second pick after yellow) = 8/23
To find the probability of either event occurring (black then yellow or yellow then black), we add the probabilities together:
P(One black and one yellow) = P(Black on first pick) * P(Yellow on second pick after black) + P(Yellow on first pick) * P(Black on second pick after yellow)
= (1/3) * (10/23) + (1/3) * (8/23)
= 10/69 + 8/69
= 18/69
= 6/23
Therefore, the probability that one bead is black and the other is yellow is 6/23.
To summarize:
A. The probability that both beads are red is 5/92.
B. The probability that one bead is black and the other is yellow is 6/23.