Question

۱۳) میخواهیم از بین 3 دانش آموز دهم رشته ریاضی و 2 دانش آموز دهم رشته تجربی یک تیم دو نفره تنیس انتخاب کنیم، چقدر احتمال دارد؟
الف) هر دو نفر از دانش آموزان دهم ریاضی باشند؟
ب) هر دو نفر هم رشته باشند؟
ج) حداکثر یک نفر رشته ریاضی باشد؟

Explanation:

⚡ Using what you learned: Combinations (Unordered Selections) · 🆕 New: Probability of Events

Step 1: Total Outcomes

Total students: \( 3 \text{ (Math)} + 2 \text{ (Science)} = 5 \text{ students} \).

Selecting a 2-person team from 5 students:

$$ n(S) = \binom{5}{2} = \frac{5 \times 4}{2} = 10 $$

Step 2: Part (a) - Both are Math students

Selecting 2 students from the 3 Math students:

$$ n(A) = \binom{3}{2} = 3 $$

Probability:

$$ P(A) = \frac{n(A)}{n(S)} = \frac{3}{10} = 0.3 $$

Step 3: Part (b) - Both are from the same major

The two students can either both be from Math or both be from Science.

Selecting 2 from 2 Science students:

$$ \binom{2}{2} = 1 $$

Total favorable outcomes:

$$ n(B) = \binom{3}{2} + \binom{2}{2} = 3 + 1 = 4 $$

Probability:

$$ P(B) = \frac{n(B)}{n(S)} = \frac{4}{10} = \frac{2}{5} = 0.4 $$

Step 4: Part (c) - At most one Math student

"At most one Math student" means either 0 Math students (both Science) or 1 Math student (and 1 Science student).

Alternatively, we can use the complement (subtracting the probability of having 2 Math students from 1):

$$ P(C) = 1 - P(\text{both Math}) $$

$$ P(C) = 1 - \frac{3}{10} = \frac{7}{10} = 0.7 $$

Answer:

• الف) \( \frac{3}{10} \) (or \( 0.3 \))
• ب) \( \frac{4}{10} = \frac{2}{5} \) (or \( 0.4 \))
• ج) \( \frac{7}{10} \) (or \( 0.7 \))