Two of these children are chosen at random.
Calculate the probability that both children are taller than 1.5 m but only one of them is talle
than 1.8 m .

Question

Two of these children are chosen at random. Calculate the probability that both children are taller than 1.5 m but only one of them is talle than 1.8 m .

UpStudy AI · Accepted Answer

To determine the probability that **both** children selected are taller than **1.5 meters**, but **only one** of them is taller than **1.8 meters**, we'll follow a combinatorial approach. Here's a step-by-step method to calculate this probability.

### **Assumptions and Definitions**

Let's assume:
- **Total number of children (N):** The total number of children from which we're selecting.
- **Number of children taller than 1.5 m (A):** This includes all children taller than 1.5 meters, regardless of whether they're taller than 1.8 meters.
- **Number of children taller than 1.8 m (B):** These children are also counted in the group taller than 1.5 meters since 1.8 m > 1.5 m. So, **B ≤ A**.

### **Objective**

We want to find the probability that:
1. **Both** selected children are taller than **1.5 m**.
2. **Only one** of them is taller than **1.8 m**.

### **Calculating the Probability**

1. **Total Ways to Choose 2 Children:**
   
   The total number of ways to select 2 children out of **N** is given by the combination formula:
   
   $$
   \text{Total combinations} = C(N, 2) = \frac{N \times (N - 1)}{2}
   $$

2. **Favorable Outcomes:**

We need to select:
   - **One child** taller than **1.8 m** (there are **B** such children).
   - **One child** taller than **1.5 m** but **not** taller than **1.8 m** (there are **A - B** such children).
   
   The number of favorable ways to do this is:
   
   $$
   \text{Favorable combinations} = C(B, 1) \times C(A - B, 1) = B \times (A - B)
   $$

3. **Probability Calculation:**

The probability (**P**) is the ratio of favorable outcomes to total possible outcomes:
   
   $$
   P = \frac{\text{Favorable combinations}}{\text{Total combinations}} = \frac{B \times (A - B)}{\frac{N \times (N - 1)}{2}} = \frac{2 \times B \times (A - B)}{N \times (N - 1)}
   $$

### **Final Formula**

$$
\boxed{P = \frac{2 \times B \times (A - B)}{N \times (N - 1)}}
$$

### **Explanation of the Formula**

- **Numerator (Favorable outcomes):** Counts all possible ways to select one child taller than 1.8 m and one child between 1.5 m and 1.8 m.
- **Denominator (Total outcomes):** Counts all possible ways to select any two children from the group.
- **Multiplying by 2:** Accounts for the fact that the pair can be selected in two different orders (Child 1 from group B and Child 2 from group A-B, or vice versa).

### **Example**

Suppose:
- **Total children (N) = 10**
- **Children taller than 1.5 m (A) = 6**
- **Children taller than 1.8 m (B) = 2**

Plugging into the formula:

$$
P = \frac{2 \times 2 \times (6 - 2)}{10 \times 9} = \frac{2 \times 2 \times 4}{90} = \frac{16}{90} = \frac{8}{45} \approx 0.178
$$

So, there's approximately a **17.8%** chance of selecting two children where both are taller than 1.5 m, but only one is taller than 1.8 m.

---

**Note:** To compute the numerical probability, you'll need the specific values for **N**, **A**, and **B**. The above method provides a general approach applicable to any such problem with given values.
The probability that both selected children are taller than 1.5 meters, with only one being taller than 1.8 meters, is calculated using the formula: $$ P = \frac{2 \times B \times (A - B)}{N \times (N - 1)} $$ Where: - $$ N $$ is the total number of children. - $$ A $$ is the number of children taller than 1.5 meters. - $$ B $$ is the number of children taller than 1.8 meters. This formula gives the probability based on the given values of $$ N $$, $$ A $$, and $$ B $$.

Two of these children are chosen at random.
Calculate the probability that both children are taller than 1.5 m but only one of them is talle
than 1.8 m .

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