The supply office for a large construction firm has three welding units of Brand A and three welding units that are not Brand A in stock. If a welding unit is requested, the probability is .7 that the request will be for Brand A. Find the probability that exactly one of the non-brands A units will be left immediately after the third Brand A is requested.
To find the probability that exactly one of the non-Brand A units will be left immediately after the third Brand A unit is requested, we can use the concept of conditional probability.
Let's break down the problem step by step:
Step 1: Calculate the probability of drawing a Brand A unit
The probability of drawing a Brand A unit is given as 0.7. Therefore, the probability of not drawing a Brand A unit is:
P(Not Brand A) = 1 - P(Brand A) = 1 - 0.7 = 0.3
Step 2: Determine the probability of drawing three Brand A units in a row
Since there are three Brand A units in stock, the probability of drawing a Brand A unit on the first request is 0.7. Since there will be two Brand A units left after the first request, the probability of drawing a Brand A unit on the second request is 0.7 again. The same applies to the third request. Therefore, the probability of drawing three Brand A units in a row is:
P(3 Brand A units) = (0.7) * (0.7) * (0.7) = 0.343
Step 3: Calculate the probability of having exactly one non-Brand A unit left
To have exactly one non-Brand A unit left, there are two scenarios:
1) The first two requests draw Brand A units, and the third request draws a non-Brand A unit.
2) The first request draws a Brand A unit, the second request draws a non-Brand A unit, and the third request draws a Brand A unit.
Let's calculate the probabilities of these two scenarios separately:
Scenario 1:
P(2 Brand A units, 1 non-Brand A unit) = P(Brand A) * P(Brand A) * P(Not Brand A) = (0.7) * (0.7) * (0.3) = 0.147
Scenario 2:
P(1 Brand A unit, 1 non-Brand A unit, 1 Brand A unit) = P(Brand A) * P(Not Brand A) * P(Brand A) = (0.7) * (0.3) * (0.7) = 0.147
Step 4: Calculate the total probability
To find the probability that exactly one non-Brand A unit will be left, we sum up the probabilities of the two scenarios:
P(exactly one non-Brand A unit left) = P(2 Brand A units, 1 non-Brand A unit) + P(1 Brand A unit, 1 non-Brand A unit, 1 Brand A unit) = 0.147 + 0.147 = 0.294
Therefore, the probability that exactly one of the non-Brand A units will be left immediately after the third Brand A is requested is 0.294 or 29.4%.