An airgun can take a maximum of 4 shots at a balloon at some distance. The probabilities of hitting the balloon at the first, second, third and fourth shot are 0.1,0.2,0.3 and 0.4 respectively. What is the probability that the balloon is hit?

Options

A. 0.6976
B. 0.6576
C. 0.786
D. none of these

Answer

To solve the problem of finding the probability that the balloon is hit at least once in 4 shots, we can follow these steps:

### Step 1: Identify the probabilities of hitting the balloon
The probabilities of hitting the balloon on each shot are given as follows:
- Probability of hitting on the 1st shot (H1) = 0.1
- Probability of hitting on the 2nd shot (H2) = 0.2
- Probability of hitting on the 3rd shot (H3) = 0.3
- Probability of hitting on the 4th shot (H4) = 0.4

### Step 2: Calculate the probabilities of missing the balloon
To find the probability of not hitting the balloon on each shot, we subtract the hitting probabilities from 1:
- Probability of missing on the 1st shot = 1 - H1 = 1 - 0.1 = 0.9
- Probability of missing on the 2nd shot = 1 - H2 = 1 - 0.2 = 0.8
- Probability of missing on the 3rd shot = 1 - H3 = 1 - 0.3 = 0.7
- Probability of missing on the 4th shot = 1 - H4 = 1 - 0.4 = 0.6

### Step 3: Calculate the probability of missing all shots
To find the probability of missing the balloon in all 4 shots, we multiply the probabilities of missing each shot:
\[ P(\text{missing all}) = P(\text{missing 1st}) \times P(\text{missing 2nd}) \times P(\text{missing 3rd}) \times P(\text{missing 4th}) \]
\[ P(\text{missing all}) = 0.9 \times 0.8 \times 0.7 \times 0.6 \]

### Step 4: Perform the multiplication
Calculating the above expression:
\[ P(\text{missing all}) = 0.9 \times 0.8 = 0.72 \]
\[ 0.72 \times 0.7 = 0.504 \]
\[ 0.504 \times 0.6 = 0.3024 \]

### Step 5: Calculate the probability of hitting at least once
The probability of hitting the balloon at least once is given by:
\[ P(\text{hitting at least once}) = 1 - P(\text{missing all}) \]
\[ P(\text{hitting at least once}) = 1 - 0.3024 = 0.6976 \]

### Final Answer
Thus, the probability that the balloon is hit at least once in 4 shots is **0.6976**.

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