5 coins are tossed together. what is the probability of getting exactly 3 heads

Answer (Detailed Solution Below)

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Formula used:

P(E) = (Favorable events)/(Total events)

Calculation

Favorable case getting two heads = 5C2 = (5 × 4)/(1 × 2) = 10

Total events = 25 = 32

 P(E) = 10/32 = 5/16

∴ Five coins are tossed the probability of getting two heads is 5/16. 

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The number of possible outcomes of each coin flip is 2 (either heads or tails.) So the probability of either a heads or a tails is 1/2.

In Chapter 2 you learned that the number of possible outcomes of several independent events is the product of the number of possible outcomes of each event individually. So the number of combinations that 2 coin flips will give you is: 2 x 2 = 4. In this case we are flipping 5 coins -- so the number of possibilities is: 2 x 2 x 2 x 2 x 2 = 32.

In case you'd like to scrutinize this point, here is the list of all possible combinations:

Coin        1    2    3    4    5
outcome 1   h    h    h    h    h  (5 h)

        2   h    h    h    h    t  (4 h)
        3   h    h    h    t    h
        4   h    h    t    h    h
        5   h    t    h    h    h
        6   t    h    h    h    h

        7   h    h    h    t    t  (3 h)
        8   h    h    t    h    t
        9   h    t    h    h    t
       10   t    h    h    h    t
       11   h    h    t    t    h
       12   h    t    h    t    h
       13   t    h    h    t    h
       14   h    t    t    h    h
       15   t    h    t    h    h
       16   t    t    h    h    h

       17   t    t    t    h    h  (2 h)
       18   t    t    h    t    h
       19   t    h    t    t    h
       20   h    t    t    t    h
       21   t    t    h    h    t
       22   t    h    t    h    t
       23   h    t    t    h    t
       24   t    h    h    t    t
       25   h    t    h    t    t
       26   h    h    t    t    t 

       27   t    t    t    t    h  (1 h)
       28   t    t    t    h    t
       29   t    t    h    t    t
       30   t    h    t    t    t
       31   h    t    t    t    t

       32   t    t    t    t    t  (0 h)

As you can count for yourself, there are 10 possible ways to get 3 heads. Thus, the probability of getting 3 heads from 5 coin flips is: 10/32, or 5/16.

The ratio of successful events A = 16 to the total number of possible combinations of a sample space S = 32 is the probability of 3 heads in 5 coin tosses. Users may refer the below solved example work with steps to learn how to find what is the probability of getting at-least 3 heads, if a coin is tossed five times or 5 coins tossed together. Users may refer this tree diagram to learn how to find all the possible combinations of sample space for flipping a coin one, two, three or four times.

Solution
Step by step workout
step 1 Find the total possible events of sample space S
S = {HHHHH, HHHHT, HHHTH, HHHTT, HHTHH, HHTHT, HHTTH, HHTTT, HTHHH, HTHHT, HTHTH, HTHTT, HTTHH, HTTHT, HTTTH, HTTTT, THHHH, THHHT, THHTH, THHTT, THTHH, THTHT, THTTH, THTTT, TTHHH, TTHHT, TTHTH, TTHTT, TTTHH, TTTHT, TTTTH, TTTTT}

S = 32

step 2 Find the expected or successful events A
A = {HHHHH, HHHHT, HHHTH, HHHTT, HHTHH, HHTHT, HHTTH, HTHHH, HTHHT, HTHTH, HTTHH, THHHH, THHHT, THHTH, THTHH, TTHHH}

A = 16

step 3 Find the probability
P(A) = Successful Events/Total Events of Sample Space
= 16/32

= 0.5
P(A) = 0.5

0.5 is the probability of getting 3 Heads in 5 tosses.

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