A bag contains 4 white beads and one black bead. Three beads are removed from the bag individually being replaced before each selection .find the probability that from the three beads there are at least two black beads
At each draw,
prob(white) = 4/5
prob(black) = 1/5
so you want 2 blacks or 3 blacks
= C(5,2) (1/5)^2 (4/5) + C(5,3) (1/5)^3
= ....
you push the calculator buttons
To find the probability of drawing at least two black beads from the three draws, we need to calculate the probability of three different scenarios: drawing two black beads and one white bead, drawing three black beads, and drawing two black beads and one black bead.
Let's start by calculating the probability of drawing two black beads and one white bead:
1. Calculate the probability of drawing a black bead: In the bag, there are 4 white beads (W) and 1 black bead (B). The probability of drawing a black bead is given by P(B) = 1 / (4 + 1) = 1/5.
2. Calculate the probability of drawing a white bead: After each draw, the bead is replaced, so the probabilities remain the same for each draw. The probability of drawing a white bead is given by P(W) = 4 / (4 + 1) = 4/5.
3. Calculate the probability of drawing two black beads and one white bead: Since we want to find the probability of drawing at least two black beads, we need to consider the order of the draws. There are three possible orders that satisfy this condition: BBW, BWB, and WBB.
- For the order BBW: P(BBW) = P(B) * P(B) * P(W) = (1/5) * (1/5) * (4/5).
- For the order BWB: P(BWB) = P(B) * P(W) * P(B) = (1/5) * (4/5) * (1/5).
- For the order WBB: P(WBB) = P(W) * P(B) * P(B) = (4/5) * (1/5) * (1/5).
4. Calculate the total probability for two black beads and one white bead: Add up the probabilities for all three order possibilities: P(2 black, 1 white) = P(BBW) + P(BWB) + P(WBB).
Now, let's calculate the probability of drawing three black beads:
1. Calculate the probability of drawing three black beads: In this case, there is only one order possibility: BBB.
- P(3 black) = P(B) * P(B) * P(B) = (1/5) * (1/5) * (1/5).
Finally, let's calculate the probability of drawing two black beads and one white bead:
1. Calculate the probability of drawing two black beads and one white bead: In this case, there are three possible order possibilities: BBW, BWB, and WBW.
- For the order BBW: P(BBW) = P(B) * P(B) * P(W) = (1/5) * (1/5) * (4/5).
- For the order BWB: P(BWB) = P(B) * P(W) * P(B) = (1/5) * (4/5) * (1/5).
- For the order WBW: P(WBW) = P(W) * P(B) * P(W) = (4/5) * (1/5) * (4/5).
2. Calculate the total probability for two black beads and one white bead: Add up the probabilities for all three order possibilities: P(2 black, 1 white) = P(BBW) + P(BWB) + P(WBW).
Finally, calculate the total probability by adding up the probabilities for all three scenarios:
P(at least 2 black beads) = P(2 black, 1 white) + P(3 black)
= P(BBW) + P(BWB) + P(WBB) + P(BBB).
Note: Make sure to perform all calculations with decimal numbers, not rounded fractions, to get precise results.