We are given that a drunkard man takes a step forward with probability 0.6 and a step backward with probability 0.4. He takes 9 steps in total, and we are asked to find the probability that he is just one step away from the initial point.
Step 1: Understand the Problem
Let’s define the steps:
A forward step is considered as +1.
A backward step is considered as -1.
The total number of steps is 9, and we need to find the probability that the drunkard is just one step away from the initial point after 9 steps. This means the difference between the number of forward and backward steps must be ±1.
Step 2: Define Variables
Let:
x be the number of forward steps.
y be the number of backward steps.
Since the drunkard takes 9 steps in total: x + y = 9
We are looking for situations where the drunkard is one step away from the initial point. This means: x - y = ±1
Step 3: Solve the System of Equations
We have two equations:
x + y = 9
x - y = ±1
Case 1: x - y = 1
From x + y = 9 and x - y = 1, we can solve for x and y:
Add the two equations: (x + y) + (x - y) = 9 + 1, so 2x = 10, thus x = 5.
Substitute x = 5 into x + y = 9, we get 5 + y = 9, so y = 4.
Thus, in this case, x = 5 and y = 4.
Case 2: x - y = -1
From x + y = 9 and x - y = -1, we can solve for x and y:
Add the two equations: (x + y) + (x - y) = 9 - 1, so 2x = 8, thus x = 4.
Substitute x = 4 into x + y = 9, we get 4 + y = 9, so y = 5.
Thus, in this case, x = 4 and y = 5.
Step 4: Probability Calculation
Now, we calculate the probability for both cases. Since the drunkard has 9 steps in total, the number of ways to choose x forward steps from 9 steps is given by the binomial coefficient C(9, x), and the probability of any specific sequence of x forward steps and y backward steps is (0.6)^x * (0.4)^y.
Case 1: x = 5, y = 4
The number of ways to choose 5 forward steps from 9 is C(9, 5), and the probability for this case is: P1 = C(9, 5) * (0.6)^5 * (0.4)^4
C(9, 5) = 9! / (5! * (9-5)!) = 126 Thus, the probability is: P1 = 126 * (0.6)^5 * (0.4)^4 ≈ 126 * 0.07776 * 0.0256 ≈ 0.247
Case 2: x = 4, y = 5
The number of ways to choose 4 forward steps from 9 is C(9, 4), and the probability for this case is: P2 = C(9, 4) * (0.6)^4 * (0.4)^5
C(9, 4) = 9! / (4! * (9-4)!) = 126 Thus, the probability is: P2 = 126 * (0.6)^4 * (0.4)^5 ≈ 126 * 0.1296 * 0.01024 ≈ 0.167
Step 5: Total Probability
The total probability that the drunkard is one step away from the initial point is the sum of the probabilities of the two cases: P_total = P1 + P2 ≈ 0.247 + 0.167 = 0.414
Thus, the probability that the drunkard is just one step away from the initial point is approximately 0.414.
Step 6: Impact of Drinking Habits on Family Life
Drinking habits can indeed have a significant impact on one’s family life. Excessive drinking often leads to problems such as neglect, communication breakdown, financial strain, and emotional stress within families. In extreme cases, it can contribute to abusive behavior or even breakdown of relationships. It's important for individuals to seek help if they find their drinking habits are negatively affecting their lives or the lives of those around them.
