Question

3. A hand of five cards is drawn simultaneously (without order or replacement) from a standard 52-card deck. Let A be the event that the hand includes four cards of the same kind, and let B be the event that at least two of the cards in the hand are of the same kind. a. Compute P(A). 15 b. Compute P(B). (5) c. Compute P(AB). 15

Answer 1

use remaining define events Sol"! In this problem we crow hand of Live Cord simultimeourly (without replacement) cil; once we draw a card we never put it back in Oleck and your pwethwe prob. we Cards. So here total = 52-Card deck the A- The hond includes 4 Covedsg same kind B = There at least two g the card in the hould are of some kind So first we have to know there are fowe types of Coved in deck ; a) Spades (c) Diamonds (6) Hearts (dl) Clubs Each suit Contains 13 Cards So now bele able to solve joovet (a) (b) (0)

(a) PCA) = P(f cards of some type and 1 is cmother + P(4 axe y Spdis, one is Diomand) Color) = 4P(4 are este Spals, one in et Hecoret) of -+P(4 cvce y Spades, one to o Club) tour + P(4 are of Club, one is + P(A are of club, one is of +PC9 Dicm., 1 spadi)t pe Diom., 1. Club). + PlaDiam, 1 Heart) + P(4 Heart, I Club) + P O Heart, I picm)t ple Heart, I spade) 13C, X2C , x "C, X OC Heart) g some is o Diamond) + PCÁ Clut; 1 pa] 1801 + X W 52 2 (1X3 CX50 ( x 79 C #с. BC x 12 x "C, X "Ci 130 t 5201 5). So 99 "CX CXC Doc 12 times - 12 x 13 x 12 x 11x10 52x51x50x99 48 x 13 =>|PCA) = 0.0086 b) PCB) = Plout g 5 at least two g Same tind)

= 1- Plout 9 5, 2 are not some) - 1-P (zare og Some, 2 differunt) + 1-{p(3 Diamond, 2.0db, 1 spad, 1, club) 7 Pla Diamond, 1 Spod, 1 Hecvet) +P(+3 Diamond, i Hecerit, i club)-t -- Like Case Ca) 1-[52 12 times Flex 13 12X11 x 13x13 52x51x49 48x97 + + 13x12x1! x13x131 52251049 48x47] 12 times 2) 12 x 13 x 12 x 11 x 13x13 20.9881 52x51x49x 48x97) PCⓇ) <0.998) C) PCA)B) Conditional sob. Theorum di P(B) P(A/B) = P(AMB) uring this define event ADB = In four of same colour at heart two are of Sams; e'l, two of some color out of 5

= |- [Pl two not schm) + P 13 not some) = pre 1 - PC two cree not some) the hat someo) ] 1 - [P(3 spade, i Hercet, i cutie PleSpade, 1 Club, Bicnnord) TP (3 spad, Hert, I Diamond) f... ] tibe pard 21- 520x 13x12x11 13x13 52x51x59 99x9% techos х + 12 times 13X 12 X 13x13x13 52x51 5099*99 t + 12 times 12% 13X12X11 X13413 12x 13x12x13x13015 52x51X50X98X99 ·0112 + 0.0096] - -0.9792 z 1-[123 =|- [0.0 PCAD) - 0.9792 52x51250x 99198] to 0.9909 = P(A/B) 0.9881