Archive image from page 449 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni07rees Year: 1819 DOCTRINE OF CHANCES. in 2 p -â¢)- i trials probability of its happening p time â c d + p + i ⢠d. In like manner the probability of its happening p time, i'ueceffively at the end of 2 p + 2 trials will depend upon it' having failed upon the whole during the firft p + 1 trials, and alfo in the p + .id trial. The probability, therefore, in thiscafe, w-aibeT-e- dd = d-cd- dd; which pd- - 2 ⢠p ⢠r - 1 â p + - S
Archive image from page 449 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni07rees Year: 1819 DOCTRINE OF CHANCES. in 2 p -â¢)- i trials probability of its happening p time â c d + p + i ⢠d. In like manner the probability of its happening p time, i'ueceffively at the end of 2 p + 2 trials will depend upon it' having failed upon the whole during the firft p + 1 trials, and alfo in the p + .id trial. The probability, therefore, in thiscafe, w-aibeT-e- dd = d-cd- dd; which pd- - 2 ⢠p ⢠r - 1 â p + - Sec. -3P- 1 3-/> 3-/-2 2 -J Let n he = ;/> then will n â /> = r â 1 . / â â zp . â r~~l . p, n â 3 /â . = r â 3 . p, &c and if a. $, y, J, &c. be put for« â p n â 2)>, fl â 3/', « â 4/> being added to c â cd + 'p + 1 â d (the probability of &c. relpcdively, the probability that the tveni happens p its happeningp times in zp + I trials) will give c â 2 c d 4- times fucceffivcly in «trials will be = c â & c d c d d dd-T â c d3 + &c. -f «. d â P .P-i y â 1 d3 - 5:c p + 2. .d - ddior the whole probability of its happening p times fucceffively in 2 j> + z trials. The probability of its happening /> times in 2/ + 3 trials may be found, by proceeding in the fame manner, to be â lâ e â - d . a + c -2ed + T+~ 3cd+P + 3-d- â ,2-3 j I Example. In 30 throws with a Gngle counter, having one ââ¢, l l-v. r- 1 ⢠. ,;:,, , A 4. , trials fide white and the ot'ner black, what are the-oddthai the The probabihty of its happening /> times m 2 p -+- 4 . ' . r , , T . . r A r___ L rf ° ' , , , white does not come up ten times : In this caw to be = 1 - c-3d .d.+ c- led +_p + 3 -d- Sdd J _ ; = f-+ed+ p + 4 .</ â 6 </ times in 4- c â 4cd + p + 4 . d â 6 ddâc â 5 cd + p +.5 ⢠d â 10 </ â lit term of the fi- gurate numbers 1, 3, 6, 10, &c. is â , the proba- is = ââ . d = ââ .
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