Probability-1

Probability Level 4

Tiles in a bag are numbered from $\left\{ 0,1,2,3,4,\ldots,98,99 \right\}.$ Two tiles with the numbers $a$ and $b$ are randomly drawn from the bag with replacement. What is the probability that the number $3^a+7^b$ has a digit equal to $8$ at the unit place?

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The answer is 0.1875.

3 solutions

Mehul Arora
Mar 14, 2016

Three cases arise:

$\bullet 3^a \equiv 7^b \equiv 9 (\mod 10)$

$\bullet 3^a \equiv 1 (\mod 10) , 7^b \equiv 7 (\mod 10)$

$\bullet 3^a \equiv 7 (\mod 10) , 7^b \equiv 1 (\mod 10)$

Case 1:

Cyclic powers of 3 are $1,3,9,7$ [because the set involves 0]

Cyclic powers of 7 are $1,7,9,3$

Thus, when $a = 2,6,10,14, \cdots 98$

and $b = 2,6,10,14, \cdots , 98$ , the units digit is 8.

Thus, probability = $\dfrac {25}{100} \times \dfrac {24}{99} = \dfrac {12}{198}$ (Because there is no replacement, and the a=b is one option.)

Case 2:

Cyclic powers of 3 are $1,3,9,7$ [because the set involves 0]

Cyclic powers of 7 are $1,7,9,3$

Thus when $a = 0,4,8, \cdots 96$

and $b = 1,5,9, \cdots, 97$

The units digit is 8. Thus probability = $\dfrac {25}{100} \times \dfrac {25}{99} = \dfrac {25}{396}$

Case 3: Again,

Cyclic powers of 3 are $1,3,9,7$ [because the set involves 0]

Cyclic powers of 7 are $1,7,9,3$

Thus when $a = 3,7,11, \cdots 99$

and $b = 1,5,9, \cdots, 97$

The units digit is 8.

Thus probability = $\dfrac {25}{100} \times \dfrac {25}{99} = \dfrac {25}{396}$

Adding the probabilities of all cases, we get $P(E) = \dfrac {37}{198} = 0.187$

I thought that the question implied that replacement occurred. With that, I got the answer of 0.1875, which was accepted as 'Correct!'. I think I remember noticing this happening before... does Brilliant round answers which have more than 3 decimal points? Perhaps this question should then be rephrased to find the probability times a 10000.

Vladimir Smith - 5 years, 3 months ago

does Brilliant round answers which have more than 3 decimal points? Yes

Pi Han Goh - 5 years, 3 months ago

Pranav Rao
Mar 16, 2016

When the question said 'selected from the same set' , I thought there is replacement as otherwise after removing one element, the set won't remain the same, right? So I got answer 3/16 =0.1875 which was accepted by Brilliant. Lucky!!

That's because both with and without replacement provide roughly equal answers. $\frac{3}{16} = 0.1875$ and $\frac{37}{198} = 0.1868686...$

Vishnu Bhagyanath - 5 years, 2 months ago

Frank Petiprin
Jan 21, 2018

import random
import os
import math
import sys
def my_range1(start, end, step):
    while start <= end:
        yield start
        start += step
#End Function Definition**
#Program Written In Pythonista using the Monte Carlo method**
beg,end,incr,grandt8 = 1,2000000,1,0
#Used for testing
#random.seed( 9006 )
for toss in my_range1(beg,end,incr ):
    a = random.randint(  0, 99  )
    b = random.randint(  0, 99  )
    num = 3**a + 7**b
    if( ( num%10 ) == 8 ):
        grandt8 += 1
print(' ')
#RATIO of grandt8 to end is the probability
print('prob ', grandt8/end, end)
input('End pgm  a b')
#** ENDPROGRAM**
#       ****THREE Program Runs****

'''prob  0.1874488 20000000
End pgm a b

prob  0.1875935 2000000
End pgm a b

prob  0.1875685 2000000
End pgm a b'''

Probability-1

The answer is 0.1875.

3 solutions

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