• 39) A slot machine at a hotel is configured so that there is a 1/ 1200 probability of winning the jackpot on any individual trial. If a guest plays the slot machine 6 times, find the probability of exactly 2 jackpots. If a guest told the hotel manager that she had hit two jackpots in 6 plays of the slot machine, would the manager be surprised.
• The slot idea started in 1891 when Sittman and Pitt in New York developed a gambling machine which is considered to be the first slot machine. Since 1891, the slots are the most using type of game in the gambling industry. When you know how do online slots work, the main question is ‘’How to calculate the probability of winning an online.

1. The Probability Of Winning Slot Machine Is 8 Binomial Key
2. The Probability Of Winning Slot Machine Is 8 Binomial Nomenclature
3. Winning Slot Machine Strategy
4. The Probability Of Winning Slot Machine Is 8 Binomial Distribution
5. The Probability Of Winning Slot Machine Is 8 Binomial Squares

First of all, we must start with the number of possible combinations. In the case of slots, it is relatively simple – just multiply the numbers of symbols on each reel. The oldest slots had, for example, 3 reels with ten different symbols on each. The total number of combinations that could appear on the panel was 1,000 (10 x 10 x 10).

The number of combinations in today’s slots is somewhat higher. If we assume five reels with 30 symbols on each, we get a total of 243,000,000 combinations.

For the binomial distribution, SD is equal to, where is the number of rounds played, is the probability of winning, and is the probability of losing. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold.

If you want to calculate your chances to win on an online slot machine, all you need is this simple equation:

Number of winning combinations / Total number of combinations

To calculate the payout of the slot machine, modify the formula a little:

Σ (winning combination_k * possible yield_k) / (Total number of combinations)

Let’s analyze a few basic slot machines. For the purposes of our article and in order to simplify the calculation, we will assume that the slot machine has only one payout line and the bet is one coin per round.

Analysis of the simplest slot machine

Let’s go back to the past and assume that the machine only has 3 reels and there is an apple, an orange, a lemon, a banana, a melon and a joker symbol on each. The individual combinations produce these winnings:

1. Three jokers win 30 coins
2. Any three fruits win 10 coins
3. Two jokers win 4 coins
4. One joker wins 1 coin

The total number of combinations is 216 (6 x 6 x 6).

Total number of winning combinations:

1. In the first case there is only one winning combination (1 x 1 x 1 = 1)
2. In the second case we have 5 winning combinations (3 times apple or 3 times orange or 3 times lemon, …) (1 x 1 x 1) x 5 = 5
3. The joker may appear on any two reels. The calculation is as follows: 1 x 1 x 5 + 1 x 5 x 1 + 5 x 1 x 1 = 15
4. The joker may appear on any reel. 1 × 5 × 5 + 5 × 1 × 5 + 5 × 5 × 1 = 75

Our simplified model thus contains 1 + 5 + 15 + 75 = 96 winning combinations. The table below shows the probability of a payout.

Calculation of payouts according to the formula

Σ (winning combination_k * possible yield_k) / (Total number of combinations)

(1 × 30 + 5 × 50 + 15 × 4 + 75 × 1)/(6 × 6 × 6) = 215/216 ≈ 0.99537

In this case, the slot machine has a payout ratio of 99.53%, which is very nice, but in a real casino, you will not find the same results. The average returns of slots online casinos will be between 94% and 98%.

The table also clearly shows how single coin wins affect payouts. If the win of each combination were equal to one coin, the winning ratio would drop to 44.4%. And that’s a very small number.

Analysis of a more complicated slot

Because the previous example was too distant from reality, let’s show you another example with higher numbers. To simplify, let’s assume again that there is only one payline, the slot machine has 3 reels and a total of 6 symbols that can appear on the panel:

The total number of combinations is 23 x 23 x 23 = 12,167.

Winning combinations with single coin returns:

1. 3x BAR, win 60 coins, number of combinations 1
2. 3x SEVEN, win 40 coins, number of combinations 3 x 1 x 1 = 3
3. 3x Cherry, win 20 coins, number of combinations 4 x 3 x 3 = 36
4. 3x Other fruit, win 10 coins, number of combinations (5 x 6 x 6) x 3 = 540
5. Cherry on two reels, win 4 coins, number of combinations 651
6. Cherry on one reel, win 1 coin, number of winning combinations 3,880

Calculation for no. 5:

Cherry, Cherry, Other: 4 x 3 x (23 – 3) = 240

Cherry, Other, Cherry: 4 x (23 – 3) x 3 = 240

Other, Cherry, Cherry: (23 – 4) x 3 x 3 = 171

Calculation for no. 6:

First reel: 4 x 20 x 20 = 1,600

Second reel 19 x 3 x 20 = 1,120

Third reel 19 x 20 x 3 = 1,120

The following table shows the amount of payout and the chance of winning for the individual combinations.

As you can see, the payout ratio is very high again at 99.721% (12,133 / 12,161). If the slot were to pay a straight win for each winning combination in the amount of 1 coin, the payout ratio would be down to 42,007%.

As in any other statistical areas, the understanding of binomial probability comes with exploring binomial distribution examples, problems,answers, and solutions from the real life.
It is not too much to say that the path of mastering statistics and data science starts with probability. And the binomial concept has its core role when it comes to defining the probability of success or failure in an experiment or survey.

On this page you will learn:

• Binomial distribution definition and formula.
• Conditions for using the formula.
• 3 examples of the binomial distribution problems and solutions.

Many real life and business situations are a pass-fail type. For example, if you flip a coin, you either get heads or tails. You either will win or lose a backgammon game. There are only two possible outcomes – success and failure, win and lose.

And the key element here also is that likelihood of the two outcomes may or may not be the same.

So, what is binomial distribution?

Let’s define it:

In simple words, a binomial distribution is the probability of a success or failure results in an experiment that is repeated a few or many times.

The prefix “bi” means two. We have only 2 possible incomes.

Binomial probability distributions are very useful in a wide range of problems, experiments, and surveys. However, how to know when to use them?

Let’s see the necessary conditions and criteria to use binomial distributions:

• Rule 1: Situation where there are only two possible mutually exclusive outcomes (for example, yes/no survey questions).
• Rule2: A fixed number of repeated experiments and trials are conducted (the process must have a clearly defined number of trials).
• Rule 3: All trials are identical and independent (identical means every trial must be performed the same way as the others; independent means that the result of one trial does not affect the results of the other subsequent trials).
• Rule: 4: The probability of success is the same in every one of the trials.

Notations for Binomial Distribution and the Mass Formula:

The Probability Of Winning Slot Machine Is 8 Binomial Key

Where:

• P is the probability of success on any trail.
• q = 1- P – the probability of failure
• n – the number of trails/experiments
• x – the number of successes, it can take the values 0, 1, 2, 3, . . . n.
• _nC_x = n!/x!(n-x) and denotes the number of combinations of n elements taken x at a time.

Assuming what the _nC_x means, we can write the above formula in this way:

Just to remind that the ! symbol after a number means it’s a factorial. The factorial of a non-negative integer x is denoted by x!. And x! is the product of all positive integers less than or equal to x. For example, 4! = 4 x 3 x 2 x 1 = 24.

Examples of binomial distribution problems:

• The number of defective/non-defective products in a production run.
• Yes/No Survey (such as asking 150 people if they watch ABC news).
• Vote counts for a candidate in an election.
• The number of successful sales calls.
• The number of male/female workers in a company

So, as we have the basis let’s see some binominal distribution examples, problems, and solutions from real life.

Example 1:

Let’s say that 80% of all business startups in the IT industry report that they generate a profit in their first year. If a sample of 10 new IT business startups is selected, find the probability that exactly seven will generate a profit in their first year.

First, do we satisfy the conditions of the binomial distribution model?

• There are only two possible mutually exclusive outcomes – to generate a profit in the first year or not (yes or no).
• There are a fixed number of trails (startups) – 10.
• The IT startups are independent and it is reasonable to assume that this is true.
• The probability of success for each startup is 0.8.

The Probability Of Winning Slot Machine Is 8 Binomial Nomenclature

We know that:

n = 10, p=0.80, q=0.20, x=7

The probability of 7 IT startups to generate a profit in their first year is:

This is equivalent to:

Interpretation/solution: There is a 20.13% probability that exactly 7 of 10 IT startups will generate a profit in their first year when the probability of profit in the first year for each startup is 80%.

And as we live in the internet ERA and there are so many online calculators available for free use, there is no need to calculate by hand.
Just use one of the online calculators for binomial distribution (for example this one).

The important points here are to know when to use the binomial formula and to know what are the values of p, q, n, and x.

Also, binomial probabilities can be computed in an Excel spreadsheet using the =BINOMDIST function.

Example 2:

Your basketball team is playing a series of 5 games against your opponent. The winner is those who wins more games (out of 5).

Let assume that your team is much more skilled and has 75% chances of winning. It means there is a 25% chance of losing.

What is the probability of your team get 3 wins?

We need to find out.

In this example:

n = 5, p=0.75, q=0.25, x=3

Let’s replace in the formula to get the answer:

Interpretation: the probability that you win 3 games is 0.264.

Example 3:

A box of candies has many different colors in it. There is a 15% chance of getting a pink candy. What is the probability that exactly 4 candies in a box are pink out of 10?

Winning Slot Machine Strategy

We have that:

n = 10, p=0.15, q=0.85, x=4

When we replace in the formula:

The Probability Of Winning Slot Machine Is 8 Binomial Distribution

Interpretation: The probability that exactly 4 candies in a box are pink is 0.04.

The above binomial distribution examples aim to help you understand better the whole idea of binomial probability.

The Probability Of Winning Slot Machine Is 8 Binomial Squares

If you need more examples in statistics and data science area, our posts descriptive statistics examples and categorical data examples might be useful for you.