Back to the cards! In poker, a flush is when all five cards are the same suit. Find the probability of being dealt a flush (when being dealt five cards).
Start by just considering clubs.

- anonymous

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- anonymous

a) What is the probability that the first card dealt is a club?
b) What is the probability that the second card dealt is a club given that the first one was a club?
c) What is the probability that the third card dealt is a club given that the first two were clubs?
d) What is the probability that the fourth card dealt is a club given that the first three were clubs?
e) What is the probability that the fifth card dealt is a club given that the first four were clubs?
f) The probability of being dealt all five clubs is the product of the above probabilities. Why is this true and what is this probability?
g) You have now found the probability of being dealt a flush in clubs. This is the same as the probability of being dealt a flush in diamonds, hearts, or spades. Then, what is the proability of being dealt a flush?

- hba

How many cards in a deck ?

- anonymous

Umm... im not sure

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## More answers

- anonymous

@mathmate

- anonymous

@Hero @radar

- anonymous

@marsss

- anonymous

There are 52 cards in a standard deck. Divide by 4 to get the number of cards of each suit.

- radar

That would mean there are 13 club cards out of 52 cards. Wouldn't that mean the probability of drawing a club is 1 out of 4?

- mathmate

Yes, drawing the first club out of the full deck of 52 cards has a probability of 1 out of 4, or 0.25.
What about the probabiliity of the second card:
- how many clubs are left (assuming first one is a club) ?
- how many cards are left in the deck?

- radar

Now on the 2nd draw there would be 13 club cards out of 51
3rd draw 12 out of 50
4th draw 11 out of 49
5th draw 10 out of 48

- mathmate

...huh?

- radar

Wouldn't that be the chances for each draw assuming he draws into a 5 card flush.
Now take the product of all those chances to get the probability.
1/4 * 13/51 * 6/25 * 11/49 * 5/24

- radar

I would guesstimate a little less than 1400 to 1

- radar

What does google say?

- anonymous

hey I'm here now

- anonymous

so a i got 1/4. what is b?

- radar

Chance according to Google is 0.003940. or about 254 to 1 so it looks like I probably erred.

- anonymous

so b is 254/1?

- anonymous

or .004?

- radar

I am afraid I have been away from those kind of problems too long. Hopfully, @mathmate will provide further assistance.

- anonymous

Do u know anyone online right now that can give me assistance now?

- anonymous

@Agent_Sniffles

- anonymous

@zepdrix

- radar

The Google solution involved a flush in any suit not just clubs.

- anonymous

huh?

- radar

The probability of a flush can occur in hearts, diamonds, spades or clubs, just as long as all cards are the same suit!

- anonymous

So what would my answers be?

- radar

You requested the probability of a flush in clubs only.

- anonymous

Yea. thats what the question said

- anonymous

how would I start answering B?

- radar

I answered that, if the first one was a club, you now have a deck of 51 cards of which 12 are clubs; 12/51 or 0.235294

- anonymous

Ok so b would then be 12/51?

- radar

Here is mathmate. hopefully shed some light on this.

- anonymous

I have (a) 1/4
(b) 12/51
now c?

- radar

Ask yourself how many cards are now in the deck, how many are clubs and figure out the probability using the method you have been taught.

- mathmate

Sorry for being away.
@radar sorry, I was just questioning in case there was a typo.
For (b)
After the first card, there are 13-1=12 clubs left out of 51.
So the probability is 12/51.
Or, using conditional probabilities:
P(1)=13/52
P(1&2)=13/52*12/51
P(2|1)=P(2&1)/P(1)=(13/52*12/51) / (13/52) = 12/51, same as before.

- anonymous

Okay I got that for (b) too!:) Im not sure how to find c now

- radar

Sorry but I have to now run, you are in good hands.

- anonymous

Would (c) be 11/51 then?

- mathmate

@schmidtdancer
The questions are made in such a way to guide you to the final answer. I suggest that after a and b have been explained and answered, it would be advantageous for you to continue the logic and post your suggested responses for verification.
What do you think?

- anonymous

? I just need clarification on how to find C. can u guide me through the steps....and ill figure it out by myself then u can check?

- anonymous

@mathmate

- anonymous

@precal

- mathmate

Sure!

- anonymous

Thanks! how do i begin c?

- anonymous

c) What is the probability that the third card dealt is a club given that the first two were clubs?

- mathmate

For the third club in a row, how many clubs are left?
and how many cards are left in the deck?

- anonymous

since there are 14 clubs in a set, then we would have 11 left right?

- anonymous

because b is 12/51? and were losing another so would c be 11/51?

- mathmate

I'll make it clear:
After the first two clubs are drawn and before we draw the third card, how many clubs remain in the deck, and how many cards total remain?

- anonymous

there are 14 clubs total in a deck... and 52 cards total in a deck... so, after two are drawn, then we have 12 clubs and 50 cards?

- anonymous

is that for b or c?

- anonymous

@mathmate ????

- anonymous

@countonme123

- mathmate

Each deck has 52 cards, divided by 4 suits gives 13 cards per suit to start with (not 14).
(a) before drawing any card, we have 13 clubs and 52 cards.
(b) given the first card drawn was a club, before drawing the second card, we have 12 clubs and 51 cards.
(c) given the first 2 cards drawn were clubs, before drawing the third card, we have how many clubs and how many cards in the deck?

- anonymous

10/49?

- anonymous

@mathmate

- mathmate

We have only drawn 2 clubs out of 13, how many left?

- anonymous

11

- mathmate

Sorry, the OS seems to be very selective in response. I cannot get to your question unless I go by your profile. When I go by "mathematics", it never responds for the past two days.

- anonymous

Its ok. but is 11 right?

- mathmate

(c)
Also, drawn two cards (clubs) out of 52, how many left?

- anonymous

50

- anonymous

so 11/52 would be c?

- anonymous

I mean 11/50!!!!

- anonymous

right @mathmate

- mathmate

Yes, that is correct for (c).
Again, your response was not updated. I had to check through you profile every time I suspect a response.
You're comfortable continuing?

- anonymous

Okay, and yes

- anonymous

How would I find (d) now @mathmate

- mathmate

Each deck has 52 cards, divided by 4 suits gives 13 cards per suit to start with (not 14).
(a) before drawing any card, we have 13 clubs and 52 cards.
(b) given the first card drawn was a club, before drawing the second card, we have 12 clubs and 51 cards.
(c) given the first 2 cards drawn were clubs, before drawing the third card, we have how many clubs and how many cards in the deck?
(d) given the first 3 cards drawn were clubs, before drawing the fourth card, we have how many clubs and how many cards in the deck?

- anonymous

so (d) is: 10/49?

- anonymous

This is what I have.... is it correct @mathmate ?
(a) 1/4
(b) 12/51
(c) 11/50
(d) 10/49
(e) 9/48

- mathmate

Looks good so far. Keep it up, you're almost there!

- anonymous

Im a little confused on f and g!

- mathmate

Have you done conditional probability before?

- anonymous

Kinda

- anonymous

When it says product, do I just multiply a-e?

- anonymous

?@mathmate

- mathmate

Yes, the numerical part is just the product of the 5 probabilities.
Give me a minute for the explanation part of (f).
Once you have the numerical probability of clubs obtained in (f), how would you propose to find (g)?

- anonymous

so f is then: 11880/23990400??

- anonymous

@mathmate

- anonymous

Then would g be the decimal value of f?

- mathmate

It's almost correct, but you need to simplify it to the simplest form. Can you do that?

- anonymous

I was gonna ask u lol, I'm not sure how to simplify it... im trying

- mathmate

No. Think of (g) is for the case where there are 4 suits instead of just clubs.
Imagine buying raffle tickets. What are the changes of winning if you bought 4 instead of one?

- anonymous

I have 495/999600 so far

- anonymous

for f..

- anonymous

33/66640

- anonymous

Is this f?? 33/66640

- mathmate

Factors that you can cancel are like 36...

- anonymous

is this right @mathmate 33/66640

- mathmate

That is correct.

- anonymous

Ok thank you! now, g?

- mathmate

Now try (g)

- anonymous

hmm?

- anonymous

g) You have now found the probability of being dealt a flush in clubs. This is the same as the probability of being dealt a flush in diamonds, hearts, or spades. Then, what is the proability of being dealt a flush?

- anonymous

im not sure what its asking for an answer....@mathmate

- mathmate

Means a flush of any of the 4 suits.
Imagine buying raffle tickets. What are the changes of winning if you bought 4 instead of one?

- mathmate

*chances

- anonymous

4 times as much?@mathmate

- anonymous

?

- mathmate

Right!
That's. I hope you are better prepared for the next question.

- anonymous

But what would g be??

- anonymous

Is the answer: 4 times as much? @mathmate

- mathmate

4 times what you got for (f).

- anonymous

so is it: .002?

- anonymous

132/266560 in fraction form <-- is this the answer @mathmate

- mathmate

I would rather put it as 4*(33/66640) =33/16660.
In probabilities, small numbers like 0.002 is better represented by fractions.

- anonymous

thx! so g @mathmate is 33/16660?

- anonymous

but thats lower than f.

- mathmate

Yes.
I am sorry it is probably painful for you as much as for me because the system does not respond (does not update). So I don't really know when you put in a response.

- anonymous

its ok but @mathmate how is g 33/16660? when f is higher than that value?

- mathmate

When the denominator is 4 times smaller, it means that the fraction is 4 times bigger. For example, 1/4 is smaller than 1/1.

- anonymous

Oh oak!! thanks for all your help

- mathmate

You're welcome! Good luck with your homework/exam! :)

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