Three men and three women are waiting to be interviewed for jobs. If they are all selected in random order find the probability of each event. All the women will be interviewed first. No man will be interviewed until at least two women have been interviewed
Well, it seems like the women are stealing all the spotlight here! Let's break it down:
First, let's calculate the total number of possible orders the six candidates can be interviewed. This can be found using the factorial function (!), denoted by the exclamation mark. The factorial of a number n is the product of all positive integers less than or equal to n.
Total possible orders = 6!
Now, in order for all the women to be interviewed first, we need to fix their order while allowing the men to be in any order. So the number of possible orders is now:
Possible orders = (3! x 3!) x 3!
Next, for no man to be interviewed until at least two women have been interviewed, we need to consider two sub-events:
1. Two women being interviewed, followed by one man being interviewed:
Here, we can fix the order of the men and allow the women to be in any order. So the number of possible orders is:
Possible orders = 3! x (3! x 3!)
2. Three women being interviewed, followed by all three men being interviewed:
Again, we can fix the order of the women and allow the men to be in any order. So the number of possible orders is:
Possible orders = (3! x 3!) x 3!
Finally, let's calculate the probability of each event by dividing the number of possible orders by the total number of possible orders:
1. Probability of all women being interviewed first:
Probability = Possible orders / Total possible orders
2. Probability of no man being interviewed until at least two women have been interviewed:
Probability = (Possible orders + Possible orders) / Total possible orders
And that's how you calculate the probability in this situation! Stay positive and don't let the women hog all the interview time!
To find the probability of each event, we first need to determine the total number of possible outcomes. Since there are 6 people waiting to be interviewed, there are 6! (6 factorial) ways to arrange them in a random order.
Now let's consider the specific events:
Event 1: All the women will be interviewed first.
To calculate the total number of outcomes where all the women are interviewed first, we need to consider that the women must be interviewed in a particular order (three women followed by three men). We can do this by fixing the positions of the women and men within that order.
First, we need to choose the positions for the women, which can be done in 3! ways (3 factorial). Then, the men can occupy the remaining positions, which can be done in 3! ways as well. Since these two choices are independent, we can multiply them together to get the total number of outcomes where all the women are interviewed first: 3! * 3! = 6 * 6 = 36.
Therefore, the probability of Event 1 is given by the ratio of the number of outcomes where all the women are interviewed first (36) to the total number of possible outcomes (6! = 720): P(Event 1) = 36/720 = 1/20.
Event 2: No man will be interviewed until at least two women have been interviewed.
To calculate the total number of outcomes where no man is interviewed until at least two women have been interviewed, we can think of it as arranging the women and men in a way that satisfies this condition.
First, let's consider the possibilities for the first two positions, which have to be women. We need to choose 2 women out of 3, which can be done in C(3, 2) = 3 ways (where C(n, r) represents the number of combinations of r items out of a set of n). Once we have placed these two women in the first two positions, we have fixed them and can move on.
Now, we need to arrange the remaining 4 people (1 woman and 3 men). Since the order in which they are placed is still not restricted, we can arrange them in 4! ways.
Therefore, the total number of outcomes where no man is interviewed until at least two women have been interviewed is given by C(3, 2) * 4! = 3 * 24 = 72.
Hence, the probability of Event 2 is given by the ratio of the number of outcomes where no man is interviewed until at least two women have been interviewed (72) to the total number of possible outcomes (6! = 720): P(Event 2) = 72/720 = 1/10.
In summary:
- The probability of all the women being interviewed first (Event 1) is 1/20.
- The probability of no man being interviewed until at least two women have been interviewed (Event 2) is 1/10.