# Two coins are randomly drawn from a coin pouch. If there are 4 dimes, 3 nickels, and a quarter, how many ways can at least 20 cents be drawn out?

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Two coins are randomly drawn from a coin pouch. If there are 4 dimes, 3 nickels, and a quarter, how many ways can at least 20 cents be drawn out?

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Step 1

The probability density function of X is obtained below:

From the given information, number of coins selected in a pouch is 2, number of dimes is 4, number of nickels is 3 and one quarter.

One quarter=25 cents

One dime=10 cents

One nickel=5 cents

The total number of coins in a pouch is,

Number of coins=4+3+1=8

Let the random variable Q be the quarter is selected,  D be the dime is selected and N be the nickel is selected.

Let S be the sample space.

S={DD,NN,NQ,DQ,ND}

Let X be the random variable that represents the total value.

The value for selecting coins nickel and nickel=5+5= 10

The value for selecting coins dime and dime=10+10=20

The value for selecting coins nickel and quarter=5+25=30

The value for selecting coins nickel and dime=5+10=15

The value for selecting coins dime and quarter=10+25=35

The number of ways to select 2 coins from 8 coins=combination(8,2)=28

P(X=10)=P(selecting coins nickel and nickel)=combination(3,2)/28=3/28

P(X=20)=P(selecting coins dime and dime)=combination(4,2)/28=6/28

P(X=30)=P(selecting coins nickel and quarter)=[combination(3,1)*combination(1,1)]/28=3/28

P(X=15)=P(selecting coins nickel and dime)=[combination(3,1)*combination(4,1)]/28=12/28

P(X=35)=P(selecting coins dime and quarter)=[combination(1,1)*combination(4,1)]/28=4/28

The probability density function of X is,

Step 2

The number of ways can at least 20 cents be drawn out is obtained below:

To obtain required number of ways add the numerator values in a probability density function of the ran...

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