Can't figure this problem out... wasn't great at the combinatorics unit, then it turns out everything carries over to probability.
five couples attend a bridge party, including mr. and mrs. smith. they are randomly assigned partnership pairs.
1. probability mr. smith will have mrs. smith has a partner?
2. probability all five couples will be partners?
thanks!
Hmmmm. Randomly assigned parteners? Does that mean Mr Smith can get Mr Jones as a partner? The problem says randomly, so I assume same gender couples are ok.
1. (1/9) if same gender couples are ok.
2) (1/9)(1/7)(1/5)(1/3) check that thinking.
Now, if same gender are not allowed..
1) 1/5
2) 1/5* 1/4*1/3 * 1/2
check that.
To solve these probability problems related to couples at a bridge party, we need to understand combinatorics principles. Let's break down each question step by step:
1. To find the probability that Mr. Smith will have Mrs. Smith as a partner, we need to calculate the probability of this specific event happening out of all the possible outcomes.
Step 1: Total Possible Outcomes
Since there are 10 people in total (5 couples), the total number of possible outcomes is the number of ways you can pair up 10 people. This can be calculated as:
10 choose 2 (10C2) = (10! / (2! * (10-2)!)) = 45.
Step 2: Desired Outcome
In this case, we want Mr. Smith to be paired with Mrs. Smith. So, they need to be selected together as a pair. Out of the 10 people, Mrs. Smith can be chosen as Mr. Smith's partner in only one way.
Step 3: Calculate Probability
The probability is the ratio of the desired outcome to the total number of possible outcomes:
Probability = desired outcome / total possible outcomes
Probability = 1 / 45
Therefore, the probability that Mr. Smith will have Mrs. Smith as a partner is 1/45.
2. To find the probability that all five couples will be partners, we need to calculate the probability of this specific event happening out of all possible outcomes.
Step 1: Total Possible Outcomes
Similar to before, the total number of possible outcomes is the number of ways you can pair up the 10 people: 45.
Step 2: Desired Outcome
For all five couples to be partners, we need to consider them as five separate pairs. Since each pair can be arranged in two ways (for example, Mr. A can be paired with Mrs. B or Mrs. B's partner), the desired outcome should be calculated as (2^5) = 32.
Step 3: Calculate Probability
The probability is the ratio of the desired outcome to the total number of possible outcomes:
Probability = desired outcome / total possible outcomes
Probability = 32 / 45
Therefore, the probability that all five couples will be partners is 32/45.