## The Probability That Exactly One Of The Four Balls Is White

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### The Probability That Exactly One Of The Four Balls Is White

A bag contains 12 white balls and 8 black balls; another contains 10 white balls and 15 black balls. If two balls are drawn, without replacemement, from each bag, find the probability that exactly one of the four balls is white.
I need working and explanation please
Chikis
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

Hello, Chikis!

A bag contains 12 white balls and 8 black balls;
another contains 10 white balls and 15 black balls.
If two balls are drawn, without replacemement, from each bag,
find the probability that exactly one of the four balls is white.

$\text{W\!e have: }\;\; \fbox{\begin{array}{c}\text{Bag 1} \\ \\ \hline \\12 W \\ \\ 8 B \end{array}} \;\;\;\; \fbox{\begin{array}{c}\text{Bag 2} \\ \\ \hline \\ \\ 10 W \\ \\ 15 B\end{array}}$

$\text{There are two possible scenarios.}$

$[1]\;WB\text{ is drawn from Box 1, }\,BB\text{ is drawn from Box 2.}$

. . $\text{The probabilities are: }\:\begin{Bmatrix}P(WB\text{, Box 1}) &=& \frac{{12\choose1}{8\choose1}}{{20\choose2}} &=& \frac{96}{190} \\ \\ \\ \\ P(BB\text{, Box 2}) &=& \frac{{15\choose2}}{{25\choose2}} &=& \frac{105}{300}\end{Bmatrix}$

. . $\text{Hence: }\:P(WB|BB) \:=\:\frac{96}{190}\,\cdot\,\frac{105}{300} \:=\:\frac{84}{475}$

$[2]\;BB\text{ is drawn from Box 1, }\,WB\text{ is drawn from Box 2.}$

. . $\text{The probabilities are: }\:\begin{Bmatrix}P(BB\text{, Box 1}) &=& \frac{{8\choose2}}{{20\choose2}} &=& \frac{28}{190} \\ \\ \\ \\ P(WB\text{, Box 2}) &=& \frac{{10\choose1}{15\choose1}}{{25\choose2}} &=& \frac{150}{300}\end{Bmatrix}$

. . $\text{Hence: }\:P(BB|WB) \:=\:\frac{28}{190}\,\cdot\,\frac{150}{300} \:=\:\frac{35}{475}$

$\text{Therefore: }\:P(\text{exactly 1 White}) \;=\;\frac{84}{475}\,+\,\frac{35}{475} \;=\;\frac{119}{475}$

soroban
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

Don't be offended please, I cannot see the working clearly.
Chikis
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

1:12W,8B ; 2:10W,15B
The White can be drawn 1st, 2nd, 3rd or 4th : so 4 different cases :

Case1: W=12/20, B=8/19, B=15/25, B=14/24: 12/20 * 8/19 * 15/25 * 14/24 = 42/475

Case2: B=8/20, W=12/19, B=15/25, B=14/24: 8/20 * 12/19 * 15/25 * 14/24 = 42/475

Case3: B=8/20, B=7/19, W=10/25, B=15/24: 8/20 * 7/19 * 10/25 * 15/24 = 7/190

Case4: B=8/20, B=7/19, B=15/25, W=10/24: 8/20 * 7/19 * 15/25 * 10/24 = 7/190

42/475 + 42/475 + 7/190 + 7/190 = 119/475 (as per Sir Soroban!)
I'm just an imagination of your figment...
Denis
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

My thanks goes to soroban and Denis, for having intrest in my question, let me work with what you have given me. I will post more replies into the thread, if I found anything confusing with the working.
Chikis
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

Bag I = 12w , 8B balls
Bag II = 10w , 15B balls

we have 4 possibilities(Condition )

1) Bag I(WB) and BagII(BB) = 12/20 * 8/19 * 15/25 * 14/24
2) Bag I(BW) and BagII(BB) = 8/20 * 12/19 * 15/25 * 14/24
3) Bag I(BB) and BagII(WB) = 8/20 * 7/19 * 10/25 * 15/24
4) Bag I(BB) and BagII(BW) = 8/20 * 7/19 * 15/25 * 10/24

Then P(Probability of getting exactly one white) = Condition 1 + Condition 2 + Condition 3 + Condition 4

M4mathematics
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

Don't confuse the OP; that solution is exactly same as the one I gave...
I'm just an imagination of your figment...
Denis
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

Thanks for the wonderful replies!
Chikis
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

If am to choose based on logical order, I assume that the two balls are removed simultanously from each of the two bags, so they are not mutually exclusive events.

FIRST BAG
total number of balls in the first bag
= 12+8
= 20
total number of white balls
= 12 white
total number of black balls
= 8
probability of picking a white ball P(w)
= 12/20
= 3/5
probability of picking a black ball P(b)
= 8/20
= 2/5

SECOND BAG
total number of balls in the second bag
= 10+15
= 25
total number of white balls
= 10
total number of black balls
= 5
probability of picking a white ball P(w)
= 10/25
= 2/5
probability of picking a black ball P(b)
= 15/25
= 3/5

|Bag 1| Bag 2 |
|bw | bw |
|wb | wb |
|ww | ww |
|ww | ww |

-> 2/5*3/4*3/5*2/4
= 36/400

-> 3/5*2/4*2/5*3/4
= 36/400

->
= 3/5*2/4*2/5*1/4
= 12/400

->
= 3/5*2/4*2/5*1/4
= 12/400

= (36+36+12+12)/400
= 96/400
= 6/25
= 0.24
That is my own working. What is your view about the working?
Chikis
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

An urn contains 8 white, 6 blue, and 9 red balls. How many ways can
6 balls be selected to meet each conditon?

a) all balls are red
b) 3 are blue, 2 are white, and 1 is red
c) 2 are blue and 4 are red
d) exactly 4 balls are white

There are 8 + 6 + 9 = 23 balls total. For the first question I tried
(9/23)*(8/22)*(7/21)*(6/20)*(5/19)*(4/18) but I think I did it wrong
this!
Kristenastewart
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
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### Re: The Probability That Exactly One Of The Four Balls Is Wh

Let me go back to drawing board using the actual number of marbles this time without reducing any fraction to the lowest term.

From the first bag
P(w) = 12/20
P(b) = 8/20
From the second bag
p(w) = 10/25
p(b) = 15/25

|Bag 1| Bag 2 |
case1|bw | bb |
case2|wb | bb |
case3|bb | wb |
case4|bb | bw |

I think have to reduce the fraction where possible to make it easier for further simplification.
In case 1 we have
8/20*12/19*15/25*14/24
= 2/5*12/19*3/5*7/12
= 504/5700 = 42/475

In case 2 we have
12/20*8/19*15/25*14/24
= 3/5*8/19*3/5*7/12
= 504/5700
= 42/475

In case 3 we have
8/20*7/19*10/25*15/24
= 2/5*7/19*2/5*15/24
420/11400 = 7/190

In case 4 we have
8/20*7/19*15/25*10/24
= 2/5*7/19*3/5*5/12
= 210/5700 = 7/190
Probability of getting excatly one white will now become
(84+84+35+35)/950
= 238/950
= 119/475
= 0.251
I hope this one is better now?
=
Chikis
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