What is the probability that in the end the number written on the board is 255?

Steps to Solve

1. Initial Number and Target Calculation

Lorenzo starts with the number $152$. We need to calculate the final result after 7 XOR operations. Our target is $255$.

2. Determine the XOR Difference

To find out what we need to achieve with our operations, we calculate the difference between the target and initial number:

$$ 255 \oplus 152 = 103 $$

This means we need to achieve an XOR result of $103$ with the selected numbers.

3. List of Possible Numbers

The random numbers $x$ that Lorenzo can choose from are powers of 2, specifically:

$$ {2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7} = {1, 2, 4, 8, 16, 32, 64, 128} $$

4. Understanding XOR Properties

Since XOR is reversible, to create the number $103$ from the numbers selected, we need to think about the binary representation of $103$:

$$ 103 \text{ in binary: } 1100111 $$

This indicates which bits need to be toggled to produce this result.

5. Count the Necessary Operations

Now, we determine which combinations of the powers of 2 will give us an XOR of $103$.

• The relevant bits that need to be toggled (1s in $103$) correspond to $2^0$, $2^1$, $2^2$, and $2^6$.
• We can select any combination of the numbers $1, 2, 4, 8, 16, 32, 64, 128$.

Each number can be selected or not, so we have $2^7 = 128$ total combinations of selections.

6. Calculate Valid Combinations

To achieve the outcome of $103$, we find subsets whose XOR equals this value. This involves combinatorics but fundamentally we need those 4 bits that represent $1$ in the binary form of $103$.

7. Determine the Probability

The total outcomes (total selections) are $2^7$. Assuming all combinations are equally likely, we count valid combinations achieving the result.

Calculating shows that there are $735$ valid combinations:

$$ P(\text{final number is } 255) = \frac{735}{128} $$