Probability of Outcome?

[jk2021]

Hi,

So I'm trying to help out with a math problem and its been a very long time since I have done word problems with probability (and when I studied them, it was very briefly). The problem is below (with an additional bonus).

Three teams of four members each must be chosen for the upcoming school challenge. The teams will draw cards from a hat with the name of each team on a card. The team names are the Tigers, the Warriors and the Yellowjackets. What is the likelihood that Tigers and Yellowjackets teams are filled before a single player is chosen for the Warriors team?

Bonus: If there were only two teams (Tigers and Warriors), what is the likelihood that the Tigers team would be chosen before a single person joined the Warriors?

So I'm guessing I would start by determining the probability of all being chosen on one team first then the remaining but I start to remember something about order mattering and I get lost. I've searched the web a fair amount and looked up calculators to give an idea but I don't feel like those are giving me the correct answer.



I guess its really knowing how to set it up and what rules matter because it has to be a specific outcome on a specific team (without going to the 3rd team) which feels maybe overly complex than I'm making it. Anyway, any guidance is greatly appreciated.

Thanks in advance!

PS. Also, the drawing from the hat doesn't happen all at once, it happens one card at a time. I guess this was clarified due to different outcomes if the drawing was simultaneous? But not sure.

 

[Dr.Peterson]

Three teams of four members each must be chosen for the upcoming school challenge. The teams will draw cards from a hat with the name of each team on a card. The team names are the Tigers, the Warriors and the Yellowjackets. What is the likelihood that Tigers and Yellowjackets teams are filled before a single player is chosen for the Warriors team?

This could have been worded better; as you point out, it should say the cards are drawn (and looked at) one at a time, and also it should be made clearer that it is not teams, but individuals, who draw cards saying which team they will be on (and that there are 12 of them).

Now, the first step in solving any problem like this is to understand what it means! To visualize it better, let's just say we have 12 cards with the initial of a team on them; the cards must say TTTTWWWWYYYY. Now we shuffle them, maybe WYTTWYWTYWTY. (This is an example of an "outcome" -- not an individual person, as you implied.) If they were drawn in this order, with a Warrior chosen first, the event we are asking for did not occur. (An "event" is a set of outcomes that are considered "successes".)

Now, what will it have to look like if the event does occur, and all the T's and Y's come before any of the W's?

One you understand what the event is, you will have to decide how to find its probability. That depends on what you can recall, or learn anew; I can't tell what you know about permutations, or about how to determine and combine probabilities of individual events. (These are two possible approaches you could use.)

It's also unclear whom you are helping, and why. Do they know any more about this than you do? Can you work together to pool your knowledge?

At any rate, please answer my question about what the event looks like, and show any further thoughts you have.

 

[pka]

Three teams of four members each must be chosen for the upcoming school challenge. The teams will draw cards from a hat with the name of each team on a card. The team names are the Tigers, the Warriors and the Yellowjackets. What is the likelihood that Tigers and Yellowjackets teams are filled before a single player is chosen for the Warriors team?

Bonus: If there were only two teams (Tigers and Warriors), what is the likelihood that the Tigers team would be chosen before a single person joined the Warriors?

In addition to the questions Prof Peterson posted can you tell us how many ways the string [MATH]AAAABBBCC[/MATH] can be rearranged?

 

[jk2021]

Hi,

So I am helping my daughter out with this, she had brief exposure in grade school but is exposed to it more now in Jr High/High school. I took some basic statistics 20+ years ago and have some recollection of the chance of the outcome. I did some further guesstimating based upon some more internet research and came up with this. Let me know if you are able if I am on the right track. Thanks for responding!

 

[Dr.Peterson]

You haven't defined what you mean by "outcome A"; your table isn't very helpful, as far as I can see. (It appears to be saying that the probability of picking Tigers is 2/3, which is wrong.) And your calculations seem to assume each choice is independent (as if the selection was "with replacement", which it isn't). Also, you are not taking into account the different orders in which cards might be chosen;. So you're going off in the wrong direction.

What I really need to know is, what particular ideas about probability is she currently being exposed to?? If I knew that, I could give a hint related to it.

In particular, pka's hint assumed she is learning something about combinatorics (counting things like combinations and permutations). That is a very useful approach to this problem, but I can't tell yet whether it would only confuse your daughter (or you).

If you only know how to multiply probabilities, you can consider the probability (without replacement) that the first person does not pick a W (8/12), then the probability that the second person does not pick a W (7/11, because there are only 11 cards left), and so on. Only the first of these is 2/3, and none are 1/3.

 

[lex]

Say the Tigers and YellowJackets cards are Red, and the Warriors cards are Blue. Then the hat contains 8 Red and 4 Blue cards. You just want the probability that the first person picks red, then the second person picks red....the eighth person picks red.

 

[Steven G]

All the outcomes are equally likely. That is, the probability of TWYTWYTWYTWY = probability of (TTWWYYTYW) =....
How many possible outcomes are there and how many of them have the four w's at the end?