Re: Probabilities and complex numbers.



novelo wrote:
> I've never seen probability at school and if someone can help me,
> you'll make my day for today! The problem is the following.
>
> A box contains 2 red balls and 3 white balls. A trail is to take one
> ball at random out of the box. Repeat this trial 4 times, without
> putting balls back to the box that have previously been taken out.
>
> 1) The probability for which two red balls are included in the set of 4
> balls taken out is = ?

Let R=red, W=white.

P(2 red balls in 4 trials)= 1 - P(not 2 R in 4 trials)
= 1- P(at most 1 R in 4 trials)-
P(more than 2 R in 4 trials)
= 1 -P(1 R in 4 trials) since there cannot be
4 W in 4 trials and there cannot be more than 2 R in 4 trials.
= 1 -P(R,W,W,W) -P(W,R,W,W)-P(W,W,W,R)

P(R,W,W,W),P(W,R,W,W) and P(W,W,W,R) are easy to compute, pls give it a
go.

> 2) Subject to the condition that the ball taken out in the first trial
> is a white ball, the "conditional probability" for which 2 red balls
> are included in the ser of 4 balls taken out is?
>

Same trick as above, only now think of the problem of "2 R in 3 trials
given first ball is W".

>
> If someone was able to explain me what is a conditinal probability,
> that will be very helpfull.
>

For probability on a discrete space like in the problem above, the
conditional probability of A given B is defined as P(A|B)=P(A and
B)/P(B), if P(B) is not zero. A conditional probability is again a
probability, however the probability of A is now changed by the
knowledge of B.

> Another one I cannot undestand...
>
> If a cubic equation if X with real coefficients x^3-ax^2+bx-4=0 has a
> complex root 1+i, then a=?, b=?
>
> How can I get both values and where do I need to put the complex root.

note that since 1+i is a root and the polynomial has real
coefficients, so its conjugate 1-i is also a root of the polynomial...
the trick is then to find the last root. let the roots be r1=1+i,
r2=1-i and r3 is unknown, then use the relations r1+r2+r3=a,
r1*r2+r2*r3+r3*r1=b and r1*r2*r3=4 to obtain r3, a and b..... simple,e
h?

.

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