Question: 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 = number of hearts observed.
(a) Show that X is a binomial random variable.
(b) Find the probability of observing fewer than 4 hearts in this game.
Answer:
The x binomial of random variable is [tex]\frac{1}{4}[/tex] and the probability of observing fewer hearts than 4 is 0.7759
Explanation:
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 = number of hearts observed.
(a) The X is a binomial random variable.
Binomial with n = 10;
p(heart) = [tex]\frac{13}{52}[/tex]
= [tex]\frac{1}{4}[/tex]
Draw results are independent because each drawn card is replaced.
(b) The probability of observing fewer than 4 hearts in this game.
P([tex]0\leq x\leq 3[/tex] )= binomcdf (10,[tex]\frac{1}{4}[/tex],3)
= 0.7759
This random variable's x binomial is [tex]\frac{1}{4}[/tex], and the probability of seeing fewer than 4 hearts is 0.7759. Suppose you're playing this game in which I pull a card, note its suit, replace it, shuffle, and continue until you've seen 10 cards.
Defining X as the amount of hearts witnessed.
For point a)
Demonstrating X is a binomial random variable.
Binomial with [tex]n = 10[/tex]
[tex]\to p(heart) = \frac{13}{52} = \frac{1}{4}[/tex]
Since each drawn card is changed, the results of such draws are independent.
For point b)
In this game, calculate your likelihood of seeing less than 4 hearts.
[tex]\to P(0\leq x \leq 3) = \binomcdf(10,\frac{1}{4},3) = 0.7759[/tex]
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