Answer :
Answer:
Step-by-step explanation:
As there are total 52 cards in a deck and we have to draw a set of 5 cards, we can use the formula of combination to find the total number of possible ways of drawing 5 cards.
Number of ways to draw 5 cards = [tex]N_T[/tex]
[tex]N_T\;=\;({}^NC_k)\\\\N_T\;=\;({}^{52}C_5)\\\\N_T\;=\;2,598,960[/tex]
(a) Assuming the cards are drawn in order (would not affect the probability). The of getting Ace, 2, 3, 4 and 5 can be obtained by multiplying the probability of getting cards below 6 (20/52) with the probability of getting 5 different cards (4 choices for each card).
[tex]P(a)\;=\;\frac{20}{52}*\frac{4}{52}*\frac{4}{51}*\frac{4}{50}*\frac{4}{49}*\frac{4}{48}\\\\P(a)\;=\; 1.3133*10^{-6}[/tex]
(b) For a straight we require our set to be in a sequence. The choices for lowest value card to produce a sequence are ace, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Hence, the number of ways are [tex]({}^{10}C_1)[/tex].
For each card we can draw from any of the 4 sets. It can be described mathematically as: [tex]({}^{4}C_1)*({}^{4}C_1)*({}^{4}C_1)*({}^{4}C_1)*({}^{4}C_1)\;=\;[({}^{4}C_1)^5][/tex]
Therefore, the total outcomes for drawing straight are:
[tex]N_S\;=\;({}^{4}C_1)*({}^{4}C_1)^5\;=\;10240[/tex]
Thus, the probability of getting a straight hand is:
[tex]P(b)\;=\;\frac{N_S}{N_T}\\\\P(b)\;=\;\frac{10240}{2598960}\\\\P(b)\;=\; 0.0039[/tex]