Respuesta :
To find the probability of an event A to occur, we use the formula:
[tex]P(A)=\frac{favorable\text{ }outcomes}{total\text{ }outcomes}[/tex]Also, the probability of two independent events, A and B, to occur at the same time is:
[tex]P(A\text{ }and\text{ }B)=P(A)\cdot P(B)[/tex]First, we need to find how many picture cards are in a deck.
The picture cards of each suit are: Jack, Queen, and King. Since there are 4 suits:
[tex]Picture\text{ }cards=3\cdot4=12[/tex]Now, for the first card dealt we have 52 total outcomes (the total cards in the deck) and the favorable outcomes are the 12 picture cards in the deck. We calculate:
[tex]P(1st\text{ }card)=\frac{12}{52}=\frac{3}{13}[/tex]For the second card dealt, now there is one less card in the deck, and also there is one picture card less:
[tex]P(2nd\text{ }card)=\frac{11}{51}[/tex]Similar to the second card, now there is one card less in the deck, and there is one picture card less:
[tex]P(3rd\text{ }card)=\frac{10}{50}=\frac{1}{5}[/tex]Finally, for the fourth card:
[tex]P(4th\text{ }card)=\frac{9}{49}[/tex]To find the probability of all these events happening at the same time, we multiply all the probabilities:
[tex]P(4\text{ }picture\text{ }cards)=\frac{3}{13}\cdot\frac{11}{51}\cdot\frac{1}{5}\cdot\frac{9}{49}=\frac{99}{54145}[/tex]Thus, the answer is:
[tex]Probability\text{ }dealt\text{ }4\text{ }picture\text{ }cards=\frac{99}{54145}\approx0.0018284[/tex]
In decimal, rounded to 6 decimal places, the probability is 0.001828.
