# Liars Dice Bidding Probability Calculation

Liars Dice is a game for two or more players.  During a hand, each player rolls a set of die (typically starting the game with 5 die per player) under a cup to shield their hand from the other players.  Each player then progressively makes ever increasing bids as to the number of die of a certain face value that are contained in everyone’s collective hands.  The bid is based on what each player can see in their hand and the number of concealed die in the other players hand.  The hand starts with one player bidding on the number of die displaying a particular face value ( i.e. there is one die displaying a face value of 3,  or there is 1 – 3).   The next player must either raise the bids quantity or face value higher (i.e. there is 1 – 4, 2 – 3’s, etc) or challenge the prior bid by calling the other player a liar.  Once a player is called a liar, everyone’s hand is revealed and the bid is evaluated.  If the bid is higher than the number of die of the particular face value, then the bidder loses and must remove a die from his or her hand. However, if the number of die of a particular face value exceeds that of the bid, then the challenger loses and must remove a die from his or her hand.  This process repeats until there is only one person with die in their hand and this person is declared the winner.

The challenging portion of this game is determining what to raise the bid to and when to challenge another player’s bid.  For each die there is a 1 in 6 chance that a particular face value will be rolled;  however, the number of die with a particular face value is difficult to estimate even with the knowledge of the face values of the die in your own hand.  Plus as the game progresses, the number of die in circulation is progressively decreased changing the odds.  It is much more likely that there are at least 2 – 6’s when there is 20 die in circulation than when there are only 2 die in circulation.  It is also much more likely that there are at least 2 – 6’s than 20 – 6’s when there is 20 die in circulation.

## Applying Excel’s BINOMDIST() Function

Excel’s BINOMDIST(x, n, p, cumulative) function returns the probability of “x” or fewer successes in “n” independent trials given a probability of success of “p”, when “cumulative” is true.  This function can be modified as follows such that it returns the probability of “x” or more successes in “n” independent trials given a probability of success of “p” when cumulative is true.  Using this equation, one may calculate the probability of “x” or more die being rolled with the same face value out of “n” total die rolled given the probability of a single die rolling a particular face value “p”.

```P(x or more successes) = 1 - BINOMDIST(x - 1, n, p, cumulative)

Where:

x = bid for number of die with a particular face value (in addition to what's in you hand)
n = number of die in circulation (not in your hand)
p = probability of a single die rolling a particular face value
cumulative = True (Returns the CDF)
```

## Liars Dice Bidding Probability (No Wilds)

The previous equation was used to calculate the probability that a particular bid is correct based on the number of die in circulation. The probability of a single die rolling a particular face value is 1 in 6, or 16.66%.  The resulting probabilities for varying bids and quantities of die in other players hands have been listed in the table below and color coded such that 0% chance is red, 50% chance is yellow, and 100% chance is green.

Example Bid (How to interpret the table):

If you are playing with 3 other people (5 die per player) then there are 15 die in your opponents hands.  Based on the chart above, there is a 47% chance that 3 or more of the 15 die in your opponents hands will have a particular face value.  If you happen to be dealt 2 – 2’s, then you might bid 5 – 2’s and you would have a 47% chance of being correct.

## Liars Dice Bidding Probability (Single Wild)

One popular variation of the game is to make a particular face value wild (traditionally the 1’s). The probability of a single die rolling a particular face value is 1 in 3, or 33.33% (due to the wilds). The resulting probabilities for varying bids and quantities of die in other players hands have been listed in the table below and have been color coded such that 0% chance is red, 50% chance is yellow, and 100% chance is green.

Example Bid (How to interpret the table):

If you are playing with 3 other people (5 die per player) then there are 15 die in your opponents hands.  Based on the chart above, there is a 60% chance that 5 or more of the 15 die in your opponents hands will have a particular face value.  If you happen to be dealt 2 – 2’s, then you might bid 7 – 2’s and you would have a 60% chance of being correct.

Before you leave don’t forget to:

• Tell Kevin that I will beat him at this game! … eventually. 🙂

# Baltimore, You’re More Likely to be Murdered Than Raped

The FBI routinely compiles crime statistics from the 260 largest cities in the United States (with a population of 100,000 or more).  By calculating the number of occurrences per capita (per 100,000 people) from those statistics one can compare and rank the probability of occurrence of the various crimes within each of  those cities. Baltimore’s crime statistics and ranking are listed below as derived from the 2015 FBI crime statistics. Generally, the highest ranking crime has been listed first; however, rape has been listed just under murder.  In reviewing Baltimore’s per capita statistics, it was noted that one is more likely to be murdered than raped in Baltimore and it seemed appropriate to display them together.

 Baltimore Rank Crime per Capita (per 100,000) #2 Murder 23 #122 Rape 22 #2 Robbery 293 #6 Violent crime 677 #12 Arson 21 #18 Aggravated assault 339 #19 Burglary 574 #20 Motor vehicle theft 387 #41 Property crime 2,276 #84 Larceny- theft 1,316

Baltimore is one of only a handful of cities where you are more likely to be murdered than raped.  Other cities include:

• Miami, Forida
• Miami Gardens, Florida
• Montgomery, Alabama
• Richmond, Virginia

It should be noted that conclusions based on any statistic should always be met with some level of skepticism.  The occurrence of this somewhat unique statistic may simply be due to crimes being under reported in those cities.

If your curious what other cities “beat out” Baltimore in the rankings for various crimes or want to see how your city measures up, the ranking of all 260 cities is included below.  You may click on the links below to jump to a specific table.

Also, before you leave don’t forget to:

# You’re a Sucker for Chasing New Powerball Jackpots

… and so am I.

While news of the astronomical Powerball jackpots are showing up everywhere, little is mentioned of the Powerball game changes on October 7, 2015 that are driving them up.  At that time the game was modified from a pick 5 of 59 white balls & 1 of 35 power balls (59/35) to a pick 5 of 69 white balls & 1 of 26 power balls (69/26) an the odds of winning went from 1 in 175,223,510 to 1 in 292,201,338 respectively.  The \$2 ticket you purchase today has less chance of winning than one purchased prior to October 7, 2015.  The changes to the game have also ensured higher jackpots given that the odds of winning were cut almost 60%.

Following a similar process described by Bill Butler here, I set out to determine what other impacts the new game rules had on the Powerball jackpot probabilities.  I plotted the probability of no one winning the jackpot, one ticket winning the jackpot, and two or more tickets winning the jackpot while varying the number of tickets purchased.  The solid lines represent the current 69/26 game and the dotted lines represent the old 59/35 game prior to the October 7, 2015 game changes.

This graphic shows that the October 7, 2015 Powerball game changes have increased the probability that there will be no jackpot winner within a particular drawing, decreased the probability that multiple players will win and share the jackpot, and increased the probability that a single player will take home the jackpot in games with higher participation (and likely higher jackpots).

Following the October 7, 2015 Powerball game changes, there will be larger jackpots and the occurrence of single jackpot winners will be more frequent.

In addition to us all being suckers for chasing these larger jackpots, I should also add that I’m a sucker that’s seeding those jackpots. Well, sort of.  I have a yearly Mega Millions subscription that plays my numbers every drawing.  Although I don’t have a subscription to Powerball (yet), I can imagine that many other people do.  Between those of us with automatic subscriptions and those that routinely purchase tickets while at gas stations, liquor stores, etc. we are regularly contributing to the growing jackpot while now purchasing fewer equivalent chances of winning. Once the jackpot increases significantly over the course of several games (due to those of us that regularly contribute and the decreased chance of winning) the jackpot begins to respond exponentially as those that do not purchase regularly begin buying tickets in mass.  The jackpot then increases to the point of being obscene.  The October 7, 2015 game changes appear to have targeted those that do not purchase tickets regularly in the hopes of converting them into regular subscribers.  If this tactic is successful, this conversion will in turn raise the initial and subsequent jackpots more quickly.

We are poised for some record setting single winner jackpots in upcoming years.