one card is selected at random from a deck of cards. determine the probability of selecting a card that is less than 10 or a heart. note that the ace is considered a low card.

the probability that the card selected is less than 10 or a heart is .
(type an integer or a simplified fraction.)

Question

one card is selected at random from a deck of cards. determine the probability of selecting a card that is less than 10 or a heart. note that the ace is considered a low card.

the probability that the card selected is less than 10 or a heart is  .
(type an integer or a simplified fraction.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Theoretical Probability", "Addition Rule for Probability", "Fraction Simplification" ], "new_concepts": [], "current_concepts": [ "Theoretical Probability", "Addition Rule for Probability", "Fraction Simplification" ] } ## Define the sample space and events $$ \begin{aligned} &\text{Total cards in a standard deck: } N(S) = 52\ &\text{Let } A = \text{selecting a card less than 10 (Ace is low, so values are Ace, 2, 3, 4, 5, 6, 7, 8, 9)}\ &\text{Let } B = \text{selecting a heart} \end{aligned} $$ ## Calculate individual and intersection probabilities $$ \begin{aligned} &N(A) = 9 \text{ cards per suit} \times 4 \text{ suits} = 36 \implies P(A) = \frac{36}{52}\ &N(B) = 13 \implies P(B) = \frac{13}{52}\ &N(A \cap B) = 9 \text{ cards (Ace through 9 of hearts)} \implies P(A \cap B) = \frac{9}{52} \end{aligned} $$ ## Apply the addition rule and simplify $$ \begin{aligned} P(A \cup B) &= P(A) + P(B) - P(A \cap B)\ &= \frac{36}{52} + \frac{13}{52} - \frac{9}{52}\ &= \frac{40}{52}\ &= \frac{10}{13} \end{aligned} $$ One card is selected at random from a deck of cards. Determine the probability of selecting a card that is less than 10 or a heart. Note that the ace is considered a low card. The probability that the card selected is less than 10 or a heart is $\frac{10}{13}$. (Type an integer or a simplified fraction.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Addition Rule for Probability" ] }

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Question

Explanation:

Define the sample space and events

Calculate individual and intersection probabilities

Apply the addition rule and simplify

Answer: