a card is selected at random from a standard deck of 52 playing cards. find the probability of each event.

(a) randomly selecting a black suit or a jack
(b) randomly selecting a heart or an 8
(c) randomly selecting a 3 or a face card

(a) the probability of randomly selecting a black suit or a jack is .
(type an integer or decimal rounded to three decimal places as needed.)

Question

a card is selected at random from a standard deck of 52 playing cards. find the probability of each event.

(a) randomly selecting a black suit or a jack
(b) randomly selecting a heart or an 8
(c) randomly selecting a 3 or a face card

(a) the probability of randomly selecting a black suit or a jack is  .
(type an integer or decimal rounded to three decimal places as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Classical Probability", "Mutually Exclusive Events" ], "new_concepts": [ "Addition Rule for Probability" ], "current_concepts": [ "Classical Probability", "Mutually Exclusive Events", "Addition Rule for Probability" ] } ## Identify the sample space Using the *Classical Probability* knowledge point $$ N(S) = 52 $$ ## Calculate probability for part (a) Using the *Classical Probability* and *Mutually Exclusive Events* knowledge points Let $B$ be selecting a black suit, and $J$ be selecting a jack. $$ \begin{aligned} &N(B) = 26,\quad N(J) = 4,\quad N(B \cap J) = 2\ &P(B \cup J) = P(B) + P(J) - P(B \cap J) = \frac{26}{52} + \frac{4}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13} \approx 0.538 \end{aligned} $$ ## Calculate probability for part (b) Using the *Classical Probability* and *Mutually Exclusive Events* knowledge points Let $H$ be selecting a heart, and $E$ be selecting an 8. $$ \begin{aligned} &N(H) = 13,\quad N(E) = 4,\quad N(H \cap E) = 1\ &P(H \cup E) = P(H) + P(E) - P(H \cap E) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} \approx 0.308 \end{aligned} $$ ## Calculate probability for part (c) Using the *Classical Probability* and *Mutually Exclusive Events* knowledge points Let $T$ be selecting a 3, and $F$ be selecting a face card. Since a 3 is not a face card, these are mutually exclusive. $$ \begin{aligned} &N(T) = 4,\quad N(F) = 12,\quad N(T \cap F) = 0\ &P(T \cup F) = P(T) + P(F) = \frac{4}{52} + \frac{12}{52} = \frac{16}{52} = \frac{4}{13} \approx 0.308 \end{aligned} $$ ## Question 1 The probability of randomly selecting a black suit or a jack is 0.538. ## Question 2 The probability of randomly selecting a heart or an 8 is 0.308. ## Question 3 The probability of randomly selecting a 3 or a face card is 0.308. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Statistics", "Addition Rule for Probability" ] }

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Question

Explanation:

Identify the sample space

Calculate probability for part (a)

Calculate probability for part (b)

Calculate probability for part (c)

Answer:

Question 1

Question 2

Question 3