Question

A man takes a step forward with a probability 0.4 and backward with a probability 0.6. The probability that at the end of eleven steps, he is one step away from starting point is?

Answer 1

Let's break down this problem step by step.
1. Define the Variables:

• Let 'F' represent a step forward.
• Let 'B' represent a step backward.
• Probability of a forward step \( P(F) = 0.4 \)
• Probability of a backward step \( P(B) = 0.6 \)
• Total number of steps = 11

2. Determine the Possible Scenarios:

• One step forward: 6 backward steps and 5 forward steps.
• One step backward: 6 forward steps and 5 backward steps.

3. Calculate the Number of Ways for Each Scenario:

• One step forward: The man needs 6 backward steps and 5 forward steps. The number of ways this can happen is given by the binomial coefficient: \[ \binom{11}{5} = \binom{11}{6} = \frac{11!}{5! \times 6!} = 462 \]
• One step backward: The man needs 6 forward steps and 5 backward steps. The number of ways this can happen is also given by the binomial coefficient: \[ \binom{11}{6} = \binom{11}{5} = \frac{11!}{6! \times 5!} = 462 \]

4. Calculate the Probability of Each Scenario:
 

• Probability of one step forward:
  \[ \binom{11}{5} \times (0.4)^5 \times (0.6)^6 \approx 462 \times 0.0007962624 \times 0.046656 \] which is approximately 0.170669.
• Probability of one step backward:
  \[ \binom{11}{6} \times (0.4)^6 \times (0.6)^5 \approx 462 \times 0.004096 \times 0.07776 \] which is approximately 0.115379.

5. Calculate the Total Probability:
 

• Total probability = 0.170669 + 0.115379
• Total Probability = 0.286048

6. Final Answer:
The probability that at the end of eleven steps, he is one step away from the starting point is approximately 0.2860.