Question

a board game uses the deck of 20 cards shown to the right. two cards are selected at random from this deck. calculate the probability of selecting a bird on the first draw and a lion on the second draw, both with and without replacement.

two cards are to be selected with replacement. determine the probability the first card selected shows a bird and the second shows a lion.
(type an integer or a simplified fraction.)

two cards are to be selected without replacement. determine the probability the first card selected shows a bird and the second shows a lion.
(type an integer or a simplified fraction.)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Theoretical Probability",
"Multiplication Rule for Independent Events",
"Probability without Replacement"
],
"new_concepts": [],
"current_concepts": [
"Theoretical Probability",
"Multiplication Rule for Independent Events",
"Probability without Replacement"
]
}
</pre_analysis>

<reasoning>

Analyze the deck composition

We first determine the total number of cards and the count of each category from the given image.

• Total cards: \(20\)
• Row 1 (Birds): \(5\) cards (numbered 1 to 5)
• Row 2 (Squirrels/Acorns): \(5\) cards (numbered 1 to 5)
• Row 3 (Frogs): \(5\) cards (numbered 1 to 5)
• Row 4 (Lions): \(5\) cards (numbered 1 to 5)

Using the Theoretical Probability knowledge point:

• Number of bird cards, \(n(\text{Bird}) = 5\)
• Number of lion cards, \(n(\text{Lion}) = 5\)
• Total number of cards, \(N = 20\)

Calculate probability with replacement

Using the Multiplication Rule for Independent Events knowledge point, the two draws are independent because the first card is replaced before the second draw.

• Probability of drawing a bird first:

\[P(\text{Bird}_1) = \frac{5}{20} = \frac{1}{4}\]

• Probability of drawing a lion second:

\[P(\text{Lion}_2) = \frac{5}{20} = \frac{1}{4}\]

• Combined probability:

\[P(\text{Bird}_1 \text{ and } \text{Lion}_2) = P(\text{Bird}_1) \times P(\text{Lion}_2) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\]

Calculate probability without replacement

Using the Probability without Replacement knowledge point, the outcome of the first draw affects the composition of the deck for the second draw.

• Probability of drawing a bird first:

\[P(\text{Bird}_1) = \frac{5}{20} = \frac{1}{4}\]

• After drawing a bird, \(19\) cards remain in the deck, of which \(5\) are still lions.
• Conditional probability of drawing a lion second:

\[P(\text{Lion}_2 \mid \text{Bird}_1) = \frac{5}{19}\]

• Combined probability:

\[P(\text{Bird}_1 \text{ and } \text{Lion}_2) = P(\text{Bird}_1) \times P(\text{Lion}_2 \mid \text{Bird}_1) = \frac{5}{20} \times \frac{5}{19} = \frac{1}{4} \times \frac{5}{19} = \frac{5}{76}\]
</reasoning>

<answer>

Question 1

Two cards are to be selected with replacement. Determine the probability the first card selected shows a bird and the second shows a lion.
<blank>\(\frac{1}{16}\)</blank>

Question 2

Two cards are to be selected without replacement. Determine the probability the first card selected shows a bird and the second shows a lion.
<blank>\(\frac{5}{76}\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Statistics",
"Probability without Replacement"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Theoretical Probability",
"Multiplication Rule for Independent Events",
"Probability without Replacement"
],
"new_concepts": [],
"current_concepts": [
"Theoretical Probability",
"Multiplication Rule for Independent Events",
"Probability without Replacement"
]
}
</pre_analysis>

<reasoning>

Analyze the deck composition

We first determine the total number of cards and the count of each category from the given image.

• Total cards: \(20\)
• Row 1 (Birds): \(5\) cards (numbered 1 to 5)
• Row 2 (Squirrels/Acorns): \(5\) cards (numbered 1 to 5)
• Row 3 (Frogs): \(5\) cards (numbered 1 to 5)
• Row 4 (Lions): \(5\) cards (numbered 1 to 5)

Using the Theoretical Probability knowledge point:

• Number of bird cards, \(n(\text{Bird}) = 5\)
• Number of lion cards, \(n(\text{Lion}) = 5\)
• Total number of cards, \(N = 20\)

Calculate probability with replacement

Using the Multiplication Rule for Independent Events knowledge point, the two draws are independent because the first card is replaced before the second draw.

• Probability of drawing a bird first:

\[P(\text{Bird}_1) = \frac{5}{20} = \frac{1}{4}\]

• Probability of drawing a lion second:

\[P(\text{Lion}_2) = \frac{5}{20} = \frac{1}{4}\]

• Combined probability:

\[P(\text{Bird}_1 \text{ and } \text{Lion}_2) = P(\text{Bird}_1) \times P(\text{Lion}_2) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\]

Calculate probability without replacement

Using the Probability without Replacement knowledge point, the outcome of the first draw affects the composition of the deck for the second draw.

• Probability of drawing a bird first:

\[P(\text{Bird}_1) = \frac{5}{20} = \frac{1}{4}\]

• After drawing a bird, \(19\) cards remain in the deck, of which \(5\) are still lions.
• Conditional probability of drawing a lion second:

\[P(\text{Lion}_2 \mid \text{Bird}_1) = \frac{5}{19}\]

• Combined probability:

\[P(\text{Bird}_1 \text{ and } \text{Lion}_2) = P(\text{Bird}_1) \times P(\text{Lion}_2 \mid \text{Bird}_1) = \frac{5}{20} \times \frac{5}{19} = \frac{1}{4} \times \frac{5}{19} = \frac{5}{76}\]
</reasoning>

<answer>

Question 1

Two cards are to be selected with replacement. Determine the probability the first card selected shows a bird and the second shows a lion.
<blank>\(\frac{1}{16}\)</blank>

Question 2

Two cards are to be selected without replacement. Determine the probability the first card selected shows a bird and the second shows a lion.
<blank>\(\frac{5}{76}\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Statistics",
"Probability without Replacement"
]
}
</post_analysis>