Problem Solving

At a motivational forum, a group of 1,500 people are separated into two groups, the Leaders and the Followers. 1,000 people are assigned to the Followers and 500 are assigned to the Leaders. Among these people, there are 75 married couples, each consisting of 1 Leader and 1 Follower. If 1 person is chosen randomly from each group, what is the probability that the two people chosen will be a married couple?

[spoiler]3/20,000[/spoiler]

There are 1000 people in one group, 500 people in the other, which means that if you choose one person from each group, there is a total of 1000*500 = 500000 possible pairs.

Of these 500000, we are interested only in the 75 married couples.

[spoiler]Probability = Desired / Total = 75/500000 = 3/20000[/spoiler]

Another way,

We know that there are 75 married couples, one person from each couple is in the leaders or the followers.

So first, what's the probability of getting a married person in the leaders
75/500 or 3/20

Then, what's the probability of finding the selected married person's spouse?
1/1000

Times the two probabilities, since both need to happen
(3/20)*(1/1000) = 3/20,000

jeffedwards wrote:Another way,

We know that there are 75 married couples, one person from each couple is in the leaders or the followers.

So first, what's the probability of getting a married person in the leaders
75/500 or 3/20

Then, what's the probability of finding the selected married person's spouse?
1/1000

Times the two probabilities, since both need to happen
(3/20)*(1/1000) = 3/20,000

thanks Jeff!

jeffedwards wrote:Another way,

We know that there are 75 married couples, one person from each couple is in the leaders or the followers.

So first, what's the probability of getting a married person in the leaders
75/500 or 3/20

Then, what's the probability of finding the selected married person's spouse?
1/1000

Times the two probabilities, since both need to happen
(3/20)*(1/1000) = 3/20,000

What if we start with choosing from followers, we again get 3/20,000 and then if we add the two cases we get 3/10,000?
I always get confused on when to add and when not

If both events need to occur simultaneously or at the same time,we need to multiply the probability.

If the events are in either A or B,then add the probability

selango wrote:If both events need to occur simultaneously or at the same time,we need to multiply the probability.

If the events are in either A or B,then add the probability

Exactly!

Here's an easy way to remember:

When you calculate the probability of MULTIPLE events, you MULTIPLY.

When you calculate the probability of ALTERNATIVE events, you ADD.

Words such as "and", "both" and "all" relate to multiple events.

Words and phrases such as "or", "at least" and "at most" relate to alternative probability.

mj78ind wrote:What if we start with choosing from followers, we again get 3/20,000 and then if we add the two cases we get 3/10,000? I always get confused on when to add and when not

Just a little thought on your logic, you could add them both together as you did, but you would need to add one more variable. By adding the two events you're saying you could either select the followers first or the leaders first, in which case you would have to multiply both by ½. So you would end up with 3/40,000 for both, then you'd add them together and get 3/20,000.

For example,

Select Followers
½
Select a married person from the followers
75/1000 or 3/40
Select that person's spouse from the leaders
1/500
Now multiply out
(1/2)*(3/40)*(1/500) = 1/40,000
Now do the same for the other group and add.

This is probably more confusion than help, but it might help.

Stuart Kovinsky wrote:

selango wrote:If both events need to occur simultaneously or at the same time,we need to multiply the probability.

If the events are in either A or B,then add the probability

Exactly!

Here's an easy way to remember:

When you calculate the probability of MULTIPLE events, you MULTIPLY.

When you calculate the probability of ALTERNATIVE events, you ADD.

Words such as "and", "both" and "all" relate to multiple events.

Words and phrases such as "or", "at least" and "at most" relate to alternative probability.

@Stuart

Thanks for the explaination!

jeffedwards wrote:

mj78ind wrote:What if we start with choosing from followers, we again get 3/20,000 and then if we add the two cases we get 3/10,000? I always get confused on when to add and when not

Just a little thought on your logic, you could add them both together as you did, but you would need to add one more variable. By adding the two events you're saying you could either select the followers first or the leaders first, in which case you would have to multiply both by ½. So you would end up with 3/40,000 for both, then you'd add them together and get 3/20,000.

For example,

Select Followers
½
Select a married person from the followers
75/1000 or 3/40
Select that person's spouse from the leaders
1/500
Now multiply out
(1/2)*(3/40)*(1/500) = 1/40,000
Now do the same for the other group and add.

This is probably more confusion than help, but it might help.

@Mr Edwards, Thanks.......... that is elegant!! I conceptually get it, and it certainly is not confusing

Cheers