Answer :
Using the theories of probability, we got that 0.0260 is the probability of choosing 4 spades and 2 cards that are not spades and 0.1202 is the probability of choosing 2 red face cards from a well shuffled deck of 52 playing cards..
We know a deck of 52 playing cards has 13 hearts, 13 clubs, 13 spades,
13 diamonds.
∴ The possibility that she chooses 4 spades and 2 cards that are not spades is,
= ([tex]^1^3C_4[/tex]×[tex]^3^9C_2[/tex])/([tex]^5^2C_6[/tex])
As from the 13 spade cards 4 are chosen, so now we are left with (52 - 4) = 48
and 2 cards that are not spades means out of 9 spades that are left after choosing 4 that is 9 will also be subtracted (48 - 9) = 39.
= (13!/(13 - 4)!×4!)×(39!/(39 - 2)!×2!)/52!/(52 - 6)!×6!
= (13!/9!×4!)×(39!/37!×2!)/(52!/46!×6!)
= (13×12×11×10)/(4×3×2)×(39×38)/2)/(52×51×50×49×48×47)/(6×5×4×3×2)
= (529815)/(20358520)
= 0.026.
Similarly, the probability of choosing 2 red face card is given by
=([tex]^6C_2[/tex]×[tex]^4^6C_4[/tex])/([tex]^5^2C_6[/tex])
=[ [(6! / (2!.4!)] × [46! / (4!.42!)] ]/ [52! / (6!.46!)]
=(15×163185)/20358520
=0.1202.
Hence, the probability of choosing 4 spades and 2 cards by carrie that are not spades is 0.026 and the probability of choosing 2 red face cards by carrie is 0.1202.
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