Problem 23 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A man takes a step forwards with probability \(0.4\) and backwards with probability \(0.6\). Find the probability that at the end of 11 steps he is one step away from the starting point. (a) \({ }^{11} C_{6}(0.4)^{6}(0.6)^{5}\) (c) \(462(0.24)^{5}\) (b) \({ }^{11} C_{6}(0.6)^{6}(0.5)^{5}\) (d) None of these

Step-by-Step Solution

Answer

(a) \( { }^{11} C_{6}(0.4)^{6}(0.6)^{5} \)

Step 1: Identify step types

Each step can be either a forward step with probability 0.4 or a backward step with probability 0.6. To be one step away from the starting point after 11 steps means the total number of forward steps minus the total number of backward steps is \(\pm 1\).

Step 2: Define the conditions

To end up 1 step away, in 11 steps he should have taken either 6 steps forward and 5 steps backward (ending at +1) or 5 steps forward and 6 steps backward (ending at -1).

Step 3: Calculate probability for 6 forwards, 5 backwards

The probability of taking 6 forward steps (and thus 5 backward steps) in 11 attempts is calculated using combination and multiplication of probabilities:\[ \binom{11}{6} (0.4)^6 (0.6)^5 \]

Step 4: Evaluate the expression for given options

Substitute in the given values:- \( \binom{11}{6} (0.4)^6 (0.6)^5 = 462 \times (0.4)^6 \times (0.6)^5 \).Check if this matches given options.

Step 5: Analyze provided answers

The option (a) \( { }^{11} C_{6}(0.4)^{6}(0.6)^{5} \) matches the derived expression. Thus, option (b) and option (c) do not match the required probability.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in mathematics that allows us to expand expressions that are raised to a power. It's exceptionally useful in probability, especially when dealing with a series of independent trials, like flipping a coin or rolling a die repeatedly. Each trial has two possible outcomes, which is where we see the relevance of the word "binomial." For instance, in a coin flip, you can get heads or tails.
The binomial theorem states that for any positive integer \( n \) and any real numbers \( a \) and \( b \):

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Here, \( \binom{n}{k} \) represents the binomial coefficient, which shows how many ways we can select \( k \) successes out of \( n \) trials. It's calculated as \( \frac{n!}{k!(n-k)!} \). This coefficient is a central part of solving problems where outcomes may differ from trial to trial as in our exercise where a man has probabilities of taking steps forward and backward.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and arrangements of objects. It is crucial when calculating the number of possible outcomes in probability problems. In our exercise, we see this in the use of the binomial coefficient \(\binom{11}{6}\), which helps us determine how many ways we can choose 6 forward steps out of 11 total steps.
There are two main functions in combinatorics often used in probability:

• Permutations: Used when the order of objects matters. For example, arranging three books on a shelf.
• Combinations: Used when the order does not matter. For example, choosing 3 books from a library.

In this case, since the order of forward and backward steps doesn't influence the final outcome, we use combinations. It simplifies calculations when we're only interested in the number of forwards versus backwards, without caring about specifically when each step was taken.

Probability Distribution

A probability distribution lists all possible outcomes of a random experiment and the likelihood of each outcome. These distributions are fundamental to understanding scenarios in which events occur unpredictably. In our context of a man walking 11 steps, we're dealing with a binomial distribution because each step is a trial with two outcomes—forward or backward.
The probability distribution in this scenario involves understanding:

• Number of Trials: Here, the trials are the steps taken (11 in total).
• Probability of Success: Defined by the probability of taking a step forward, which is 0.4.
• Probability of Failure: Defined as the probability of taking a backward step, which is 0.6.

The binomial distribution's probability mass function is given by:

• \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\)

For our problem, we calculate probabilities based on getting exactly 6 forward steps or 5 forward steps and apply these to compute the probability of being one step away from the starting point.