SOLUTION: Recall that there are 4 suits - spades, hearts, clubs, and diamonds - in a standard deck of playing cards. Suppose you play a game in which you draw a card, record the suit, replac

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Question 1209507: Recall that there are 4 suits - spades, hearts, clubs, and diamonds - in a standard deck of playing cards. Suppose you play a game in which you draw a card, record the suit, replace it, shuffle, and repeat until you have observed 10 cards. Define X = numbers of spades observed.

Find the probability of observing 3 or fewer spades in this game. Round to three decimal places.
Found 3 solutions by mccravyedwin, ikleyn, math_tutor2020:
Answer by mccravyedwin(407)   (Show Source): You can put this solution on YOUR website!
Probability of getting a spade = 1/4

That's a cumulative binomial distribution problem

Got a TI-84?

2nd, vars, scroll to binomcdf enter 

trials:10
p:1/4
x value:3
Paste

Scroll to Paste

enter

see binomcdf(10,1/4,3)

enter

0.7758750914 round to 0.776

Edwin
Answer by ikleyn(52793)   (Show Source): You can put this solution on YOUR website!
.
Recall that there are 4 suits - spades, hearts, clubs, and diamonds - in a standard deck
of playing cards. Suppose you play a game in which you draw a card, record the suit,
replace it, shuffle, and repeat until you have observed 10 cards. Define X = numbers of spades observed.
Find the probability of observing 3 or fewer spades in this game. Round to three decimal places.
~~~~~~~~~~~~~~~~~~~~~~~~~~ 

In this game, there are 4 suits, and every suit has the same number of card, 13,
so the total number of cards in a standard deck is 4*13 = 52.


At each step, this game returns us to the same initial condition, due to replacing and 
shuffling the cards.


Therefore, this game is a typical binomial experiment.  At each trial, we have a spade 
with the probability 1/4 = 0.25  or any other suit card with the probability 3/4 = 0.75.


So, they want you calculate the probability of having success (having a spade) 3 or fever 
times of 10 trials.


Use the standard formula for the binomial probability

    P(X <= 3) = P(0) + P(1) + P(2) + P(3)


where  P(k) = ,   are binomial coefficients   = .


Now calculate using a calculator

     P(0) =  =  = 0.056313515;

     P(1) =  =   = 0.187711716;

     P(2) =  =   = 0.281567574;

     P(3) =  =  = 0.250282288.


Finally, add the number and get 

    P(X <= 3) = 0.056313515 + 0.187711716 + 0.281567574 + 0.250282288 = 0.775875092.


Round it to 3 decimals.  So,  P(X <= 3) = 0.776  (rounded).    ANSWER


You may check this result by using a convenient online binomial calculator at

https://stattrek.com/online-calculator/binomial.aspx

At this point, the problem is solved completely.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

If your teacher lets you use a spreadsheet, then you can quickly find the answer using the BinomDist function.

The template is
=BinomDist(x,n,p,c)
x = number of successes
n = number of trials
p = probability of success
c = cumulative flag

In this case x = 3, n = 10, p = 1/4 = 0.25
c = 1 indicates that we're adding binomial probability values from x = 0 to x = 3

Therefore the full command to type in would be:
=BinomDist(3,10,0.25,1)
This leads to this approximate value 0.775875
which rounds to the final answer 0.776

Side note: Don't forget the equal sign up front of the BinomDist command.
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