Hey there,
I have been faced with this question and I am a little stuck:
An urn initially has one red ball. Persephone uses a device to select n blue balls with
probability (e^−x) x^n)/n!, for n = 0, 1, 2, . . . , and add them to the urn. She then selects
one ball at random from the urn. Show that the probability that she selects the red ball
is (1 − (e^-x) )/ x.
Regards, Seany.
Original Post
P(red selected)
ReplyDelete= sum n from 0 to infinity of P(n blue balls added and red selected)
= sum n from 0 to infinity of P(n blue balls)P(red selected|n blue balls added)
= sum n from 0 to infinity of [(e^−x)(x^n)/n!] [1/(n+1)]
= sum n from 0 to infinity of (e^−x)(x^n)/(n+1)!
= sum m from 1 to infinity of (e^−x)(x^(m−1))/m!, using m = n+1
= [(e^−x)/x] sum m from 1 to infinity of (x^m)/m!
= [(e^−x)/x] [(sum m from 0 to infinity of (x^m)/m!) − (x^0)/0!]
= [(e^−x)/x] [e^x − 1]
= (1 − e^−x)/x.