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A popular gambling game that would be played in the casino is blackjack. It is sometimes reffered to as 21

1.The basic properties of probabilitity are:

1. Each outcome has a non- negative probability. One can never have a probability below zero as events that would certainly not occur would be represented by a probability of 0. This is the case for blackjack, as the chances of having a card already in a dealers hand before shuffling them again and distributing, is zero.

2. The sum of probability must be =1. When all the cards needed for a given win are presented, their probability would add up to one. For black jack the objective of the game is to beat the dealer. If your cards total higher than the dealer's cards without going over 21 you win, this would imply that while the dealer keeps dealing and you have a cards less than him, the more he goes on dealing and you keep narrowing the gap without passing the 21 total, you narrow that gap by increasing the probability. When you surpass the dealer without going over the 21 mark, you have already increased your chance to the maximum probability 1 of winning.

3. The probability of an event is always equal to the sum of probability of its element. The case of black jack as a game, it would be in order to say that each card that the dealer gives, carries a probability of either an individual winning or losing to the casino. As the dealer deals each card, the probability at any given time of the play would be equivalent to the cards already dealt on the play.

2) venn diagrams for mutually exclusive,dependent and independent events

events is said to be mutually exclusive if they cannot both happen at the same time. For example turning your head right and left, it cannot both happen at the same time.

MUTUALLY EXCLUSIVE VENN

A B

if both A and B are events, and are mutually exclusive this is how we would represent them.

In the case of blackjack once the dealer has all the aces on his hand you cannot be having it with you, making its use to be mutually exclusive, as you will be having something different. Aces and kings are mutually exclusive.

Dependent events;

An event is said to be dependent if its outcome is affected by previous outcomes.in blackjack dependent event is demonstrate in the card type that the dealer is about to deal. That is, what the dealer is to deal to the player is dependent on the already dealt cards. Since he cannot deal already dealt cards, what he deals is highly dependent on what he has distributed already.

DEPENDENT EVENTS VENN DIAGRAM

Independent events

An event is said to be an independent event if it is unaffected by the previous events that happened. For blackjack as a game in a casino, ones win or lose does not determine the outcome of your next game. In blackjack a win maybe preceded by loss or vice versa. The outcome of one play does not determine the outcome of the next, the casino may have subsequent loses or wins or the player may face the same fate without the outcomes being dependent on anything.

DEPENDENT EVENTS VENN DIAGRAM

3 Considering the balances of probabilities, that is a normal distribution, the chances of my “play” coming up in a game varies according to the bell shaped normal distribution. Since in a normal distribution the middle part is the average, we can use to determine the standard deviation as it is the number of units to the left or right of this average. We know that 68% of that 68% of outcomes will fall within 1 standard deviation while 95% will fall within 2-standard deviation and 99% will fall within 3 standard deviations.

The chances of my “Play” coming up Is proportional to these standard deviations. It is mostly assumed by mathematicians that for blackjack game, crunching numbers and putting into account the rules blackjack, standard deviation is mostly around 1.14

It is with these that I can use to see the odds of my play coming up. The standard deviation represented in the figure below.

The respective percentages also represent the exact area under the normal distribution curve.1 standard deviation being 68% of the area, 2 standard deviation being 95% of the area while 3 standard deviation finally being 99% of the area under the curve.

If I was to bet on $20,and check my chances of losing it, this would imply that for me to calculate z-score,  I will have to first calculate the expected loss,$1 being the average for the casino. In this case it would be 20-1=19. If the number of hands was 15 then our standard deviation would be given by sd/Ѵn   where n=15

 

That would be 1.14*Ѵ15  =4.42

 

Therefore the z score would be giveb by Z= (X-µ)/ sd/Ѵn  

 

                                                                                Z= (20-1)/4.42

Z=4.3

4. Apart from “Gaming” there are several business problems that could be solved using normal distribution and z scores. For example we could have studies on management intelligence of managers within an organization questioned and put through a test to determine the suitability and check if they are effective or there was need to replace them to enhance efficiency in service delivery. This is done through conducting management aptitude test with the already expected aptitude mean given and standard deviation. It is from this measure that the scores for these managers will be evaluated normally using a normal distribution assuming all conditions for a normal distribution are met.

The z- scores would be calculated by dividing the difference between the expected aptitude average and the average gotten by the respective managers by the standard deviation.

If Let X represent the aptitude scores average with μ as the expected average and σ as the standard deviation of the expected score, We will use the normal transformation to solve these problems as illustrated below,

Z= (X- μ)/ σ

Where Z is the Z score.

AnB

B

B

A