## Probability

For students of class 11-12 (age 16+)
Abdul Muntakim Rafi
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### Probability

1. Six pennies are flipped. What is the probability of getting
a. two heads and four tails?
b. at least three heads?
c. at most two tails?

2. A bag contains one red marble and one white marble. Marbles
are drawn and replaced in a bag before the next draw.

a. What is the probability that a red marble will be drawn two
times in a row?
b. In ten draws, what is the probability that a red marble will
be drawn at least five times.

What's your answer?
And if u solve this without combinatorics,please post your solution...
Man himself is the master of his fate...

amlansaha
Posts: 100
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### Re: Probability

i don't know whether my solution includes combi or not. i am posting my solutions.
অম্লান সাহা

sakibtanvir
Posts: 188
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### Re: Probability

Amlan da,Combi should be included to solve those correctly.As instance,I am solving the first one.
We can get 2 heads and 4 tails in$\frac{6!}{2!4!}$ways.So the probability is $(\frac{1}{2})^{6}(\frac{6!}{2!4!})=\frac{15}{64}$
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.

sakibtanvir
Posts: 188
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### Re: Probability

And look at question 1(b).It says the word at least.
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.

amlansaha
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### Re: Probability

sakibtanvir wrote:And look at question 1(b).It says the word at least.
bro, i have seen the term "at least". look @ my solution. i have calculated the probabilities for the possible 3 occurance
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sakibtanvir
Posts: 188
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### Re: Probability

We can have AT LEAST three heads in ,$C(6,3)+C(6,4)+C(6,5)+1$ways.So it is not the same what u figured out,bro.
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.

nafistiham
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### Re: Probability

all that we must know for probability is GIVEN POSSIBILITES / ALL POSSIBILITIES
1.a according to combi there are $2^6$ possibilities.among them, the given one is one.so, the probability is
$\frac{1}{2^6}$

1.b for at least three heads, we don't need to think about the other three.so the possibility will be
$\frac{1}{2^3}$

1.c at most 2 heads is at least 4 tails.so, like the upper one the probability is
$\frac{1}{2^4}$
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

sakibtanvir
Posts: 188
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### Re: Probability

Should not we use the " OR law " ?????
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.

nafistiham
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### Re: Probability

what is the 'OR law' Google or Wikipedia doesn't know about it.
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

sakibtanvir
Posts: 188
Joined: Mon Jan 09, 2012 6:52 pm
Location: 24.4333°N 90.7833°E

### Re: Probability

It is a Traditional name so google cannot find it
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.