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The answer is 067.
It takes an even number of steps for the object to reach (2,2), so the number of steps the object may have taken is either 4 or 6 .
If the object took 4 steps, then it must have gone two steps [tex]$\mathrm{N}$[/tex] and two steps E, in some permutation. There are [tex]$\frac{4 !}{2 ! 2 !}=6$[/tex] ways for these four steps of occuring, and the probability is [tex]$\frac{6}{4^{4}}$[/tex].
If the object took 6 steps, then it must have gone two steps N and two steps E, and an additional pair of moves that would cancel out, either N / S or W/E. The sequences N, N, N, E, E, S can be permuted in [tex]$\frac{6 !}{3 ! 2 ! 1 !}=60$[/tex] ways. However, if the first four steps of the sequence are N, N, E, E in some permutation, it would have already reached the point (2,2) in four moves. There are [tex]$\frac{4 !}{2 ! 2 !}$[/tex] ways to order those four steps and [tex]$2 !$[/tex] ways to determine the order of the remaining two steps, for a total of 12 sequences that we have to exclude. This gives [tex]$60-12=48$[/tex] sequences of steps. There are the same number of sequences for the steps N, N, E, E, E, W, so the probability here is [tex]$\frac{2 \times 48}{4^{6}}$[/tex].
The total probability is [tex]$\frac{6}{4^{4}}+\frac{96}{4^{6}}=\frac{3}{64}$[/tex], and [tex]$m+n=067$[/tex].
What is probability?
- Probability is synonymous with possibility. It is a mathematical discipline that deals with the occurrence of a random event. The value ranges from zero to one.
- Probability has been introduced in mathematics to predict the likelihood of occurrences occurring. Probability is defined as the degree to which something is likely to occur.
- This is the fundamental probability theory, which is also utilized in probability distribution, in which you will learn about the possible results of a random experiment.
- To determine the likelihood of a particular event occurring, we must first determine the total number of alternative possibilities.
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