If you roll five dice, what are the odds of rolling five 5's?

Question

David Armstrong · Accepted Answer

Hmm..

1 Dice: chance of 1 in 6, #rarr 1/6.# 2 dice: chance of 1 in 6 for each, so: #rarr 1/6 . 1/6 = 1/36# For 2 dice: 1 in #6^2# = 1 in 36; Therefore for 5 dice: 1 in #6^5# = 1 in 776 ...

Nora Amin · Answer

The odds in favour are #1:7775#.

Assuming all five dice outcomes are independent, we can rephrase the question as, "If we roll one die five times, what is the probability of getting a 5 on each roll?" Since there is one "successful" outcome (rolling a 5) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6), the probability of rolling a 5 on one roll is #P("roll one 5")=1/6.# Here's where the independence comes in. Since we're treating the 5 dice as independent, the probability of rolling five 5's is the same as the probability of rolling one 5, five times. As such, we can multiply #P("roll one 5")# by itself 5 times, and it will give us the same answer as #P("roll five 5's").# #P("roll five 5's")= [P("roll one 5")]^5# #color(white)(P("roll five 5's"))= (1/6)^5# #color(white)(P("roll five 5's"))= 1/7776# Finally, since this question asked for the odds of rolling five 5's (and not the probability of it), we take the ratio #(P("rolling five 5's"))/(P("NOT rolling five 5's"))# which gives #(1//7776)/(7775//7776)=1/7775# Since this is an odds value, it is often written as #1:7775#. This says that for every 1 success, we should expect 7775 failures.