that these events are independent and hence by multiplying their probability together. Being independent would be equivalent to picking with replacement, any pair of people in the world, not just in a room. In short can be multiplied by itself times, which gives us Since this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is And for the group of 23 people, the probability of sharing is Poisson approximation Applying the Poisson approximation for the binomial on the group of 23 people, so The result is over 50% as previous descriptions. This approximation is the same as the one above based on the Taylor expansion that uses . Square approximation A good rule of thumb which can be used for mental calculation is the relation which can also be written as which works well for probabilities less than or equal to . In these equations, is the number of days in a year. For instance, to estimate the number of people required for a chance of a shared birthday, we get Which is not too far from the correct answer of 23. Approximation of number of people This can also be approximated using the following formula for the number of people necessary to have at least a chance of matching: This is a result of the good approximation that an event with probability will have a chance of occurring at least once if it is repeated times. Probability table {| class="wikitable" style="white-space:nowrap;" |- ! rowspan="2" | length of hex string ! rowspan="2" | no. ofbits() ! rowspan="2" | hash spacesize() ! colspan="10" | Number of hashed elements such that probability of at least one hash collision ≥ |- ! = ! = ! = ! = ! = ! = 0.001 ! = 0.01 ! = 0.25 ! = 0.50 ! = 0.75 |- align="center" | bgcolor="#F2F2F2" | 8 | bgcolor="#F2F2F2" | 32 | bgcolor="#F2F2F2" | | 2 | 2 | 2 | 2.9 | 93 | | | | | |- align="center" | bgcolor="#F2F2F2" | (10) | bgcolor="#F2F2F2" | (40) | bgcolor="#F2F2F2" | () | 2 | 2 | 2 | 47 | | | | | | |- align="center" | bgcolor="#F2F2F2" | (12) | bgcolor="#F2F2F2" | (48) | bgcolor="#F2F2F2" | () | 2 | 2 | 24 | | | | | | | |- align="center" | bgcolor="#F2F2F2"